The blowup shown by Xia doesn't rely on point particles. It works just the same even for "small spheres". I didn't really make that clear. Your thought is in g

The blowup shown by Xia doesn't rely on point particles. It works just the same even for "small spheres". I didn't really make that clear. Your thought is in good company though. Lots of physicists over the years have worked to remove point particles from the classical theories. This issue is that these fixes tend to end up fixing nothing (like with Newtonian gravity), or producing nonsense and contradictions elsewhere. A famous result in electragnetism, if we replace point charges with tiny charged spheres, we get solutions where particles accelerate exponentially without any external forces. Not at all obvious, but what the hell, right? Anyway, to take a card from your deck and muse haphazardly, my guess is that these problems are extremely hard to avoid when dealing with a continuum. There is some interesting work on discrete theories, but I'm not all that informed about them.