What justifies regularization with a high-momentum cutoff? Before renormalizing a perturbative series, during the regularization step when we insert a high-momentum UV cutoff, what justifies this step given that it's only formal and does not have a physical meaning to it?
 A: This answer is inspired by renormalization group ideas and the modern notion that all quantum field theories should be understood as an effective field theory. In spite of this, I'll attempt to keep the answer as independent from previous knowledge about these topics as possible.
When we are performing loop computations in QFT, we'll often stumble across the divergences due to the loop integrals. However, notice that when computing this integrals, we are assuming that the theory we are working with holds up to arbitrarily high energies. However, we are not sure about it. In fact, there is a lot of reason for us to doubt this: the theory we are considering in our calculations was crafted for the energies we've probed experimentally. It is quite ambitious to assume it to hold in much larger momentum scales. In particular, we certainly know that no quantum field theory should hold up to the Planck scale, for then we'd surely need to consider the effects of gravity.
Furthermore, we see experimentally that the calculations should not diverge. When we make experiments, we notice that the values are finite and well-behaved, even though in principle the calculations are blowing up. What can we conclude from these facts?
Probably we're screwing up by assuming the theory to hold for arbitrarily large momentum scales.
As I mentioned, we are always screwing up by doing this. In the very least, we should take gravity into account. Hence, to fix this problem, what we decide to do is to say "Okay, so let us say our theory only holds up to this scale $\Lambda$. It certainly can't hold up to infinite momentum, so we'll let it have a maximum scale."
We then proceed to solve the issue by just acknowledging that we were treating the theory wrong to begin with. We remove the issues coming from the bits where the theory certainly does not hold and fix the remaining free parameters experimentally.
For completion, the idea of the renormalization group follows this spirit. We integrate out the high-energy degrees of freedom that shouldn't matter and use this concept to renormalize the theory.
Another possible answer, but which I found less interesting, is that it is indeed just a formal step. Regularization allows you to see what is the divergent part of the expressions, which then allows you to renormalize by cancelling them appropriately.
