WARNING I wrote this answer a long time ago with very little idea what I was talking about. I can't delete it because it's been accepted, but I can't stand behind most of the content.
This is a very long answer and I hope it'll be helpful in some way. SPC isn't my area, but I think these comments are general enough that they apply here.
I'd argue that the most-oft-cited advantage -- the ability to incorporate prior beliefs -- is a weak advantage applied/empirical fields. That's because you need to quantify your prior. Even if I can say "well, level z is definitely implausible," I can't for the life of me tell you what should happen below z. Unless authors start publishing their raw data in droves, my best guesses for priors are conditional moments taken from previous work that may or may not have been fitted under similar conditions to the ones you're facing.
Basically, Bayesian techniques (at least on a conceptual level) are excellent for when you have a strong assumption/idea/model and want to take it to data, then see how wrong or not wrong you turn out to be. But often you are not looking to see whether you're right about one particular model for your business process; more likely you have no model, and are looking to see what your process is going to do. You do not want to push your conclusions around, you want your data to push your conclusions. If you have enough data, that's what will happen anyway, but in that case why bother with the prior? Perhaps that's overly skeptical and risk-averse, but I've never heard of an optimistic businessman that was also successful. There is no way to quantify your uncertainty about your own beliefs, and you would rather not run the risk of being overconfident in the wrong thing. So you set an uninformative prior and the advantage disappears.
This is interesting in the SPC case because unlike in, say, digital marketing, your business processes aren't forever in an unpredictable state of flux. My impression is that business processes tend to change deliberately and incrementally. That is, you have a long time to build up good, safe priors. But recall that priors are all about propagating uncertainty. Subjectivity aside, Bayesianism has the advantage that it objectively propagates uncertainty across deeply-nested data generating processes. That, to me, is really what Bayesian statistics is good for. And if you're looking for reliability of your process well beyond the 1-in-20 "significance" cutoff, it seems like you would want to account for as much uncertainty as possible.
So where are the Bayesian models? First off, they're hard to implement. To put it bluntly, I can teach OLS to a mechanical engineer in 15 minutes and have him cranking out regressions and t-tests in Matlab in another 5. To use Bayes, I first need to decide what kind of model I'm fitting, and then see if there's a ready-made library for it in a language someone at my company knows. If not, I have to use BUGS or Stan. And then I have to run simulations to get even a basic answer, and that takes about 15 minutes on an 8-core i7 machine. So much for rapid prototyping. And second off, by the time you get an answer, you've spent two hours of coding and waiting, only to get the same result as you could have with frequentist random effects with clustered standard errors. Maybe this is all presumptuous and wrongheaded and I don't understand SPC at all. But I see it in academia and in for-profit social science constantly, and I'd be surprised if things were different in other fields.
I liken Bayesianism to a very high-quality chef knife, a stockpot, and a sautee pan; frequentism is like a kitchen full of As-Seen-On-TV tools like banana slicers and pasta pots with holes in the lid for easy draining. If you're a practiced cook with lots of experience in the kitchen--indeed, in your own kitchen of substantive knowledge, which is clean and organized and you know where everything is located--you can do amazing things with your small selection of elegant, high-quality tools. Or, you can use a bunch of different little ad-hoc* tools, that require zero skill to use, to make a meal that's simple, really not half bad, and has a couple basic flavors that get the point across. You just got home from the data mines and you're hungry for results; which cook are you?
*Bayes is just as ad-hoc, but less transparently so. How much wine goes in your coq au vin? No idea, you eyeball it because you're a pro. Or, you can't tell the difference between a Pinot Grigio and a Pinot Noir but the first recipe on Epicurious said to use 2 cups of the red one so that's what you're going to do. Which one is more "ad-hoc?"
What you seem to be missing is the early history. You can check the paper by Fienberg (2006) When Did Bayesian Inference Become "Bayesian"?. First, he notices that Thomas Bayes was the first one who suggested using a uniform prior:
In current statistical language, Bayes' paper introduces a uniform
prior distribution on the binomial parameter, $\theta$, reasoning by
analogy with a "billiard table" and drawing on the form of the
marginal distribution of the binomial random variable, and not on the
principle of "insufficient reason," as many others have claimed.
Pierre Simon Laplace was the next person to discuss it:
Laplace also articulated, more clearly than Bayes, his argument for
the choice of a uniform prior distribution, arguing that the posterior
distribution of the parameter $\theta$ should be proportional to what
we now call the likelihood of the data, i.e.,
$$ f(\theta\mid x_1,x_2,\dots,x_n) \propto f(x_1,x_2,\dots,x_n\mid\theta) $$
We now understand that this implies that the prior distribution for
$\theta$ is uniform, although in general, of course, the prior may not
exist.
Moreover Carl Friedrich Gauss also referred to using an uninformative prior, as noted by David and Edwards (2001) in their book Annotated Readings in the History of Statistics:
Gauss uses an ad hoc Bayesian-type argument to show that the posterior
density of $h$ is proportional to the likelihood (in modern
terminology):
$$ f(h|x) \propto f(x|h) $$
where he has assumed $h$ to be uniformly distributed over $[0,
\infty)$. Gauss mentions neither Bayes nor Laplace, although the
latter had popularized this approach since Laplace (1774).
and as Fienberg (2006) notices, "inverse probability" (and what follows, using uniform priors) was popular at the turn of the 19th century
[...] Thus, in retrospect, it shouldn't be surprising to see inverse
probability as the method of choice of the great English statisticians
of the turn of the century, such as Edgeworth and Pearson. For
example, Edgeworth (49) gave one of the earliest derivations of what
we now know as Student's $t$-distribution, the posterior distribution
of the mean $\mu$ of a normal distribution given uniform prior
distributions on $\mu$ and $h =\sigma^{-1}$ [...]
The early history of the Bayesian approach is also reviewed by Stigler (1986) in his book The history of statistics: The measurement of uncertainty before 1900.
In your short review you also do not seem to mention Ronald Aylmer Fisher (again quoted after Fienberg, 2006):
Fisher moved away from the inverse methods and towards his own
approach to inference he called the "likelihood," a concept he claimed
was distinct from probability. But Fisher's progression in this regard
was slow. Stigler (164) has pointed out that, in an unpublished
manuscript dating from 1916, Fisher didn't distinguish between
likelihood and inverse probability with a flat prior, even though when
he later made the distinction he claimed to have understood it at this
time.
Jaynes (1986) provided his own short review paper Bayesian Methods: General Background. An Introductory Tutorial that you could check, but it does not focus on uninformative priors. Moreover, as noted by AdamO, you should definitely read The Epic Story of Maximum Likelihood by Stigler (2007).
It is also worth mentioning that there is no such thing as an "uninformative prior", so many authors prefer talking about "vague priors", or "weekly informative priors".
A theoretical review is provided by Kass and Wasserman (1996) in The selection of prior distributions by formal rules, who go into greater detail about choosing priors, with extended discussion of usage of uninformative priors.
Best Answer
So do I. But notice that there's a major ambiguity in calling something subjective.
Subjectivity (both senses)
Subjective can mean (at least) one of
Bayesianism is subjective in the second sense because it is always offering a way to update beliefs represented by probability distributions by conditioning on information. (Note that whether those beliefs are beliefs that some subject actually has or just beliefs that a subject could have is irrelevant to deciding whether it is 'subjective'.)
Actually, if a prior represents your personal belief about something then you almost certainly didn't choose it at any more than you chose most of your beliefs. And if it represents somebody's beliefs then it can be a more or less accurate representation of those beliefs, so ironically there will be a rather 'objective' fact about how well it represents them.
One could, though this doesn't tend to generalize very smoothly to continuous domains. Also, arguably it's impossible to be flat or 'indifferent' in all parameterisations at once (though I've never been quite sure why you'd want to be).
So how might we evaluate this claim?
I suggest that in the second second sense of subjective: it's mostly correct. And in the first sense of subjective: it's probably false.
Frequentism as subjective (second sense)
Some historical detail is helpful to map the issues
For Neyman and Pearson there is only inductive behaviour not inductive inference and all statistical evaluation works with long run sampling properties of estimators. (Hence alpha and power analysis, but not p values). That's pretty unsubjective in both senses.
Indeed it's possible, and I think quite reasonable, to argue along these lines that Frequentism is actually not an inference framework at all but rather a collection of evaluation criteria for all possible inference procedures that emphasises their behaviour in repeated application. Simple examples would be consistency, unbiasedness, etc. This makes it obviously unsubjective in sense 2. However, it also risks being subjective in sense 1 when we have to decide what to do when those crteria do not apply (e.g. when there isn't an unbiased estimator to be had) or when they apply but contradict.
Fisher offered a less unsubjective Frequentism that is interesting. For Fisher, there is such a thing as inductive inference, in the sense that a subject, the scientist, makes inferences on the basis of a data analysis, done by the statistician. (Hence p-values but not alpha and power analysis). However, the decisions about how to behave, whether to carry on with research etc. are made by the scientist on the basis of her understanding of domain theory, not by the statistician applying the inference paradigm. Because of this Fisherian division of labour, both the subjectiveness (sense 2) and the individual subject (sense 1) sit on the science side, not the statistical side.
Legalistically speaking, the Fisher's Frequentism is subjective. It's just that the subject who is subjective is not the statistician.
There are various syntheses of these available, both the barely coherent mix of these two you find in applied statistics textbooks and more nuanced versions, e.g. the 'Error Statistics' pushed by Deborah Mayo. This latter is pretty unsubjective in sense 2, but highly subjective in sense 1, because the researcher has to use scientific judgement - Fisher style - to figure out what error probabilities matter and shoudl be tested.
Frequentism as subjective (first sense)
So is Frequentism less subjective in the first sense? It depends. Any inference procedure can be riddled with idiosyncracies as actually applied. So perhaps it's more useful to ask whether Frequentism encourages a less subjective (first sense) approach? I doubt it - I think the self conscious application of subjective (second sense) methods leads to less subjective (first sense) outcomes, but it can be argued either way.
Assume for a moment that subjectiveness (first sense) sneaks into an analysis via 'choices'. Bayesianism does seem to involve more 'choices'. In the simplest case the choices tally up as: one set of potentially idiosyncratic assumptions for the Frequentist (the Likelihood function or equivalent) and two sets for the Bayesian (the Likelihood and a prior over the unknowns).
However, Bayesians know they're being subjective (in the second sense) about all these choices so they are liable to be more self conscious about the implications which should lead to less subjectiveness (in the first sense).
In contrast, if one looks up a test in a big book of tests, then one could get the feeling that the result is less subjective (first sense), but arguably that's a result of substituting some other subject's understanding of the problem for one's own. It's not clear that one has gotten less subjective this way, but it might feel that way. I think most would agree that that's unhelpful.