The issue that the linear predictor can take the parameter outside its admissable range is real, but not limited to this case (common examples are seen when using the identity link with Poisson or Gamma GLMs). If the data stay away from the problem area that shouldn't necessarily pose any real difficulty.
However, the two link functions don't correspond exactly, and so they literally fit different models (unless $p$ is very small throughout, in which case there's no real distinction in practice). As such, for particular applications, it's quite possible that one link function is more suitable than another, at least over the range where data are observed.
Further, in some cases, ease of interpretation may be more useful than quality of fit; if the log link fits with some theoretical understanding, for example, it may be preferable.
However, you're quite right that it's not at all difficult to convert predictions of $\text{logit}(p)$ into predictions of $p$ or $\log(p)$ so if the main reason seems to be discomfort with the $\text{logit}$ function it would seem a somewhat poor reason to avoid it. On the other hand if one were to choose log because you expected the conditional expectation of the response to be in that form, or because you wanted to explicitly model the log-mean, then it would make sense to do that.
Cons of an identity link in the case of the Poisson regression are:
- As you have mentioned, it can produce out-of-range predictions.
- You may get weird errors and warnings when attempting to fit the model, because the link permits lambda to be less than 0, but the Poisson distribution is not defined for such values.
- As Poisson regression assumes that the mean and variance are the same, when you change the link you are also changing assumptions about the variance. My experience has been that this last point is most telling.
But, ultimately this is an empirical question. Fit both models. Perform whatever checks you like. If the identity link has a lower AIC, and does as well or better on all your other checks, then run with the identity link.
In the case of the logit model vs the linear probability model (i.e., what you refer to as the identity link), the situation is a lot more straightforward. Except for some very exotic cases in econometrics (which you will find if you do a search), the logit model is better: it makes fewer assumptions and is what most people use. Using the linear probability model in its place would verge on being perverse.
As regards interpreting the models, if you are using R, there are two great packages that will do all the heavy lifting: effects, which is super easy to use, and zelig, which is harder to use but great if you want to make predictions.
Best Answer
The log link is a valid link function for the binomial family, but as you point out, its use may lead to numerical/convergence problems. So there is no guarantee that it will work well in any given case, although it often does.
There are multiple examples on this site, for instance Why isn't it 'wrong' to use a log link instead of a logit one when doing GLM with a binomial family? (really a dup!), What to Do When a Log-binomial Model's Convergence Fails and two simple examples that both also contain R code: Manipulating Binomial Distribution and Confidence interval on binomial effect size
If interest is in relative risk not in odds ratios this seems a natural way to go, see Relative Risk Regression in Medical Research: Models, Contrasts, Estimators, and Algorithms. But, as the question alludes to, there is multiple problems, not the least numerical, with fitting a binomial model with log link. There is now a dedicated R package on CRAN with multiple special algorithms trying to solve this numerical problems, see logbin: An R Package for Relative Risk Regression Using the Log-Binomial Model.