It didn’t pan out because of a poor treatment of self-reference. Correct the treatment of self-reference, then the problems get resolved. (I don’t agree with the whole of Logicism, but I think it’s much closer than any other mainstream attempt to ground mathematics).
“ Infinite sequences, sums, continuity, and all of these arguments do not require infinities to make sense of them. That’s just a piece of notation used to show the intuitive idea. Sequences, sums, and all other infinite things converge if, for every finite number, you can get “close” to the thing you claim is the solution.”
This is peculiar indeed. So you’re saying that Zeno was right? Motion is impossible, because people would only logically get “arbitrarily close” to their destination and would never arrive?
The treatment of limits is precisely the problem; when applied to the real-world, we get paradoxes — which can only be resolved by rejecting the infinite divisibility of physical reality.
Indeed, to save parts of calculus, we need to treat mathematics from a strict finitist perspective. Whether or not this is the “mainstream accepted opinion” is irrelevant. Experts have a tendency of being horrendously incorrect — while all in agreement with each other.