Full disclosure: I don't work in particles but have solved Feynman integrals for class work.

Renormalization is indeed 'black magic'. But I don't think it is due to our limited mathematical knowledge.

In my experience in mathematical physics, singularities appear when you have incomplete modeling ansatz, and so in this sense renormalization would just be a 'hack' that is necessary because of writing down imperfect equations. In other words, formulating the problem a different (and closer to truth) way would eliminate these infinities.

I know some extensions of the standard model (string theory, in particular) do not have cutoff scales. But that is largely beyond my knowledge base.

It sounds murky because we were being silly the first time around. The physically relavant quantity was always the sum of all those contributions and we decided to (arbitrarily, in the grand scheme of physics) split things up under some scheme. So, in hindsight, the positive and negative infinities were artificially manufactured by our segregation scheme... there's not much to be bothered about in that regard.

That might still mean that there's more to be understood regarding notions of infinity or the convergence of amplitudes in quantum field theory.

As an example (not sure if applicable), you could do some calculation and get an alternating series as the result:

R = 1 - 1/2 + 1/3 - 1/4 ...

If you "explain" the positive part as being contribution from field A, and the negative part as a contribution from field B, both would be infinite, but the reality might be that splitting them is nonsensical.

Well... one infinity that pops up is solved by regulators. This might be one that you get when you try to understand the casimir force. In this case, you impose an cutoff at some scale to make the expression finite, but a function of the cutoff. You send the cutoff to infinity and the expression goes to infinity.

Now the way you calculate with this is that most of the time when you try to calculate a quantity that you can measure with an instrument (in the lab or whatever), you find that the cutoff cancels. In other words, the value the cutoff took never actually mattered to begin with.

When people say that the infinities cancel or whatever it's a bit misleading. In nature we don't really know what this cutoff is. We think it should be there though! It's just sorta where the theory breaks down. Tony Zee explains this using a bedspring as an analogy. If you probe the springiness of a bedspring with a bowling ball, it works fine. It behaves as a continuous sheet. If you throw a ball bearing at it though, the bearing might fall through or hit a wire and shoot off at a weird angle. The cutoff in this case is the distance between springs or the spring size or something. Our models basically make this continuum analogy. This is NOT to say that the fabric of space time is pixelated or something like that. We have NO idea what the structure is. We just know that if we don't look too close, it works like this. The wonderful thing though is that this model is useful. There's numerous ways to build experiments that measure quantities independent of this cutoff to verify the theory. Understanding what goes on at higher energies/shorter length scales (which are effectively the same thing) is precisely why particle physics is called high energy physics.