You might say that an inverse-square force is infinite when r=0, but I would respond that classical electromagnetism doesn't really allow point charges, but only finite-density charge distributions. This probably means we should be a little bit more careful about using point charges in all our examples, but they're useful approximations if we don't think too hard about it. But point charges can't exist because they would result in infinite electric fields, which results in infinite energy density, and if you integrate the energy over any finite volume containing a point charge, you get infinite energy.
So I certainly wasn't the first one with that idea. The Feynman Lectures, volume II chapter 8, does the same integral, gets an infinite result, and concludes:
We must conclude that the idea of locating the energy in the field is inconsistent with the assumption of the existence of point charges. One way out of the difficulty would be to say that elementary charges, such as an electron, are not points but are really small distributions of charge. Alternatively, we could say that there is something wrong in our theory of electricity at very small distances, or with the idea of the local conservation of energy. There are difficulties with either point of view. These difficulties have never been overcome; they exist to this day.
I'm going to go with choice B ("there is something wrong in our theory of electricity at very small distances"), where "our theory of electricity" means classical electromagnetism. In quantum mechanics, even "point charges" aren't really localized at a point. But then I wonder what Feynman meant when he said "these difficulties ... exist to this day", since he himself was one of the main people responsible for quantum electrodynamics. So I'll have to learn QED one day and get back to you.