Technically nothing has "size" in the sense that you are intuitively defining it to be. All matter in the universe is collections of point-like (but also wave-like) particles without defined shape. It'd be totally fair to throw the critique back at you.
I don't think it would - for the obvious reason that when we talk about the size of, say, a desk, there is never a concern about its meaning, and we rarely, if ever, think about the object itself in terms of waves or particles. Protons etc., on the other hand, is a totally different matter, and since everyday notions cannot be trivially applied to such objects it becomes important to be clear concerning the meaning we choose to give to the word.
Technically, the point-like properties come out of the mathematics used, not the experimental measured. The problem with point-like is that this brings singularities to the fore when we do not measure singularities.
Singularities makes the mathematics simpler (an approximation to the reality being worked with). Keep in mind that QM and all of its associated trappings are approximations (useful but still approximations). This is no different to the use of mathematics in any other field.
The point-like properties don't just come out of the mathematics used. You can fire electrons at each other and they scatter as if point particles so that aspect of them is an experimental observation also.
Particles absolutely interact as points as well as as waves. Neither model captures their behavior accurately. They are a wave-particle thingie and that’s just what reality is.
