First of all, reversing the subject and verb isn't really going to help you. (You're not actually doing that—what you're doing is removing the subject.) In the sense that you're using it, the difference between who and that as relative pronouns has more to do with attributing consciousness to something than anything else.
What you propose may be helpful in one sense, but not in another. I can imagine somebody saying in disgust, "That went to the store?" and pointing at a person. The meaning will be understandable.
But in the verbal that, with the accompanying finger pointing and tone of disgust, there is an implied judgment that the person in question is somehow less deserving of "personhood" than other people. By using that rather than who (or he, she, or another "friendly" personal pronoun), it is an intended slight.
Which brings things back to the commonly assumed difference between who and that, where the former is used for a subject with a consciousness and the latter for one without. There is no need for your reprhasing when it's still the same distinction being made anyway. (And, aside from putting explicit emphasis on that to flag its use as a relative pronoun, asking "That went to the store?" is ungrammatical.)
So, yes. More commonly we use who when talking about people. Or about pets. Or even about inanimate objects (a car or a ship) in which (or "in whom," as the case may be) we've invested some level of personhood. In general (and discussion of non-gender pronouns aside), if we say "she" or "he" when talking about someone or something, then we use who; otherwise, we use that.
But while it's less common, there is no rule that says we cannot use that as a relative pronoun when talking about a person. It's not actually ungrammatical. In fact, you might deliberately use that while discussing the "concept" of people (nameless, random people) while still using who when discussing specific, named people.
Here's something else interesting (although somewhat unrelated): we can actually use whose when discussing inanimate objects. For example, there's nothing wrong with the sentence "The house whose rooms are filled with junk." (Although it would normally not be phrased that way.) But, at the same time (except in the context of a horror book or movie), we would not say "The house who sat empty."
Which actually means that although we're always able to substitute that for who (not that we necessarily want to), we are unable to always substitute who for that . . .