When we remove abstract numbers from a hypothetical scenario, on the other hand, nothing about the physical world seems to be affected (since, inasmuch as they are causally inert, abstract numbers are in no way responsible for the way our world is).

I can come up with possible worlds without quarks (in a vague, non-specific way). I have no idea what it means to "remove abstract numbers from a hypothetical scenario". I don't think abstract objects have modal variation which is closely related to their (not) being causal. But in so far as mathematics posits abstract entities and mathematics is explanatory than I don't think there is anything mysterious about the sense in which abstract objects are explanatory.

Abstract objects play a similar role in current physical theories to that which luminiferous aether used to play. The problem with aether isn't just that it was theoretically dispensable; it was that, even if we weren't smart enough to figure out how to reformulate our theories without assuming aether, it would still be obvious that the theoretical successes that actually motivated us to form such theories would have arisen in exactly the same way even if there were no aether. Aether doesn't predict aether-theories like ours, because our aether theory is not based on empirical evidence of aether.

I disagree. I think the problem with aether is entirely just that it was theoretically dispensable. And I think the sentences that follow that are just a way of saying "aether was theoretically dispensable".

Modern-day platonists try to make their posits appear 'metaphysically innocent' by depriving them of causal roles, but in the process they do away with the only features that could have given us positive reasons to believe such things.

Their utility in our explanations is sufficient reason to believe they exist even if their role in those explanations is not causal. Your string theory comparison doesn't sound like a successful scientific theory.

What do you mean by "work the same way"?

As in we can't develop models of possible worlds in which mathematics works differently. This has nothing to do with the abilities of hypothetical mathematicians.