In dimensional regularization, we introduce an arbitrary energy scale $\mu$. Naively, it plays the role of another parameter of the theory that needs to be fixed experimentally, but actually it is not since the RG-flow equations allow a change in $\mu$ to be compensated by a change in renormalized couplings.
The RG-flow is then to be analyzed, and based on its behavior in the UV we can either say that in the UV limit our theory reaches a fixed point and therefore is equivalent to some conformal field theory (like in QCD), or some couplings blow up in which case perturbative expansions make no sense anymore (like in QED).
My question is: why can't we just fix some $\mu$ and go from there?
In cut-off regularization scheme, for example, it is obvious that $\Lambda$ defines a "bounding energy scale" beyond which the regularized theory can not work (since the corresponding degrees of freedom simply do not exist). It is therefore natural to seek a higher $\Lambda$ whenever our current $\Lambda$ is not good enough, i.e. whenever we need to explain some higher-energy phenomena.
In contrast, in dimensional regularization we have already taken the $d \rightarrow 4$ limit, so the theory with $\mu$ and renormalized couplings evaluated at $\mu$ is already exact (well, at least perturbatively) and I can not imagine why one would need a higher $\mu$ after all.
Whilst I completely understand that in both schemes the resulting RG-flow equations are the same and define a beautiful and concise mathematical structure, I would nevertheless like to understand this conceptual point.
UPDATE: I also understand that $\Lambda$ plays a role of regularizer in the cut-off scheme, just like $d$ does in dimensional regularization (and not $\mu$). But it does not answer my question: why can't we simply take some $\mu$ and forget about the RG flow?
UPDATE 2: in case the answer to my question turns out to be "sure, why not, take some $\mu$ and go from there" $-$ then I have another one. How come that our theory can have different regimes in the UV and in the IR if I could just take some $\mu$? Maybe it has something to do with the validity of perturbative expansions?