Dimensional regularization (i.e., dim-reg) is a method to regulate divergent integrals. Instead of working in $4$ dimensions where loop integrals are divergent you can work in $4-\epsilon$ dimensions. This trick enables you to pick out the divergent part of the integral, as using a cutoff does. However, it treats all divergences equally so you can't differentiate between a quadratic and logarithmic divergence using dim-reg. All it really does is hide the fine-tuning, not fix the problem.
As an example lets do the mass renormalization of $\phi^4$ theory. The diagram gives,
\begin{equation}
\int \frac{ - i \lambda }{ 2} \frac{ i }{ \ell ^2 - m ^2 + i \epsilon } \frac{ d ^4 \ell }{ (2\pi)^4 } = \lim _{ \epsilon \rightarrow 0 }\frac{ - i \lambda }{ 2} \frac{ - i }{ 16 \pi ^2 } \left( \frac{ 2 }{ \epsilon } + \log 4 \pi - \log m ^2 - \gamma \right)
\end{equation}
where I have used the ``master formula'' in the back of Peskin and Schoeder, pg. A.44 (note that this $ \epsilon $ doesn't have anything to do with the $ \epsilon $ in the propagator). This gives a mass renormalization of
\begin{equation}
\delta m ^2 = \lim _{ \epsilon \rightarrow 0 } \frac{ \lambda }{ 32 \pi ^2 } \left( \frac{ 2 }{ \epsilon } + \log 4 \pi - \log m ^2 - \gamma \right)
\end{equation}
Keeping only the divergent part:
\begin{equation}
\delta m ^2 = \lim _{ \epsilon \rightarrow 0 } \frac{ \lambda }{ 16 \pi ^2 } \frac{ 1 }{ \epsilon }
\end{equation}
This is the same result as the one you arrived at above, but uses a different regulator. You regulated your integral using a cut-off, I did using dim-reg. The mass correction diverges as $ \sim \frac{1}{ \epsilon }$. This is where the sensitivity to the UV physics is stored.
A cutoff, which is a dimensionful number, tells you something very physical, the scale of new physics. The $\epsilon$ is unphysical, just a useful parameter.
With a cutoff, depending on how badly your divergence is, you will get different scaling with the cutoff; it will be either logarithmic, quadratic, or quartic (which has real physical significance, namely, how sensitive the result is tothe high energy physics). However, dim-reg regulated integrals always diverge the same way, like $ \frac{1}{ \epsilon } $. Dim-reg doesn't care how your integral diverges. It can be a logarithmically divergent integral but using dim-reg you will still get a $ \frac{1}{ \epsilon }$ dependence. The reason for this is that $ \epsilon $ is not a physical quantity here. Its just a useful trick to regulate the integrals.
Since dim-reg hides the type of divergences that you have, people like to say that dim-reg solves the fine-tuning problem, because by using it you don't get to see how badly your divergence is. This viewpoint is clearly flawed since the quadratic divergences are still there, they just appear to be on the same footing as logarithmic divergences when you use dim-reg.
In short the fine-tuning problem isn't really fixed using dim-reg but if you use it then you can pretend the problem isn't here. This is by no means a solution to the fine-tuning, unless someone develops an intuition for why dim-reg is the ``correct'' way to regulate your integrals, i.e., a physical meaning for $ \epsilon $ (which its safe to say there isn't one).
This is of course a very subtle problem, and I will only scratch the surface here. I think that the best reference for this question is the discussion by B. Delamotte in arxiv:0702.365, Section 2.6- "Perturbative renormalizability, RG flows, continuum limit, asymptotic freedom and all that..."
Although the two approaches seem completely different, they are in essence equivalent, in the sense that, as long as the perturbative expansion makes sense, they will give the same results : same fixed points, critical exponents, and so on.
Beyond one-loop, the two methods seem completely different : in Wilson's approach, one generates new interactions in the lagrangian ($\phi^6$ etc.), which are needed to get the fixed points at order $\epsilon^2$ for instance. In the QFT RG, one never generate new interactions in the lagrangian, everything in captured in $g_R$, which is used to generate all vertex functions (corresponding to $\langle\phi^6\rangle$ and so on). In fact, what the perturbative RG is doing in practice is to project the RG flow onto one specific RG trajectory (called L in the reference above), which can be parametrized by the flow of $m^2_R$ and $g_R$ only (thus describing the flow of all other interactions in terms of $m^2_R$ and $g_R$).
As I said in a previous answer (see below) :
In the Wilsonian approach, one starts from the microscopic scale $\Lambda$ and looks at what's going at smaller energy, whereas in the "standard" approach, one fixes than macroscopic scale and sends $\Lambda\to\infty$ in order to effectively probe smaller and smaller energy scales.
See also Divergent bare parameters/couplings: what is the physical meaning of it? Do this have any relation with wilson's renormalization group approach? and Why do we expect our theories to be independent of cutoffs?
Best Answer
Dimension regularization (dim-reg) is not very intuitive. You could say MS is not a very physical renormalization scheme. There are however several ways in which $\mu$ is connected to an actual physical energy scale in applications:
$\mu$ is arbitrary in general, however, in calculations you usually get logarithms of the form $\log (\frac{\mu}{M})$ where $M$ is some energy scale in your problem. Could be a momentum transfer for example. If you want your perturbative corrections to be small, you better choose $\mu \sim M$ otherwise the logs would be large and your perturbative correction would not be small. This is mainly the reason why $\mu$ is usually tought of as an energy scale in the problem, even though in principle it is arbitrary. If you follow this prescription for choosing $\mu$, you will find that it really is true that at high momentum transfer the QCD 2->2 scattering becomes weaker.
In problems with several interesting scales this leads to a problem, as you get several logarithms, say $\log(\frac{\mu}{m})$ and $\log(\frac{\mu}{M})$, with say $m \ll M$. In this case you cannot choose a $\mu$ such that all logarithms are small. To solve problems of this kind with dim-reg, Effective Field Theory techniques are needed. That is you first construct an EFT valid for momenta smaller than $M$ and matching small momentum S matrix elements between the theories. For definiteness, let's say $M$ is some heavy particle mass. In this case you would match the S matrix elements for the light particle between the full theory and the EFT witout heavy fields at some scale, say $\mu \sim M$, implementing the decoupling of the heavy particle BY HAND. The MS scheme does not satisfy the decoupling theorem, but you can put it in by hand. Similarly as with the former case, you match the theories at $\mu$ of order $M$ to avoid large logs in the matching.
In both cases, you put in the "interpretation" of $\mu$ by hand, to make your life easier and make perturbative correction actually small. In this sense $\mu$ in applications is usually connected to some physical scale, even though in principle it could be arbitrary.