For your "pre-Hilbert" space, I never saw that either but likely my guess at what you mean would be correct.

Can go ahead and take the completion (maybe the same as what used to be called the "normal completion"), but the result will have a lot of functions that are not differentiable and, thus, not quantum mechanics wave functions. So, I would be reluctant to take the resulting actual Hilbert space as what physics was talking about.

If we are going to talk seriously about a subject as serious as quantum mechanics, e.g., to try to understand what is known and maybe to try to clean up what is commonly called its many loose ends and maybe make progress in some applications, e.g., condensed matter physics, super conductivity, maybe even something as common as just why a grid of wire reflects radar waves, maybe connecting with gravity, dark energy, the first second, the Casimir effect, Hawking radiation, and I would include entanglement, then IMHO we need to be as clear as possible on the definitions, theorems, proofs, passing between physics and pure math, etc. After all, often in the research, we are trying to replace some of what is accepted or find what in the past was not discovered. When I see sloppy, I'm not making progress but looking at some at least some little research problems.