It’d be ideal if we could settle on a few sizes - kind of like how we have AA, AAA, C, D, etc. batteries. One can be for such mopeds, one larger for cars and some smaller ones to fill various otherwise empty spaces in a car.

This is precisely where we're going to get fucked, though, because the modern pathological mindset of every tech company now is to try to build their own proprietary walled fiefdom to try to lock in ~~suckers~~ ~~recurring revenue sources~~ customers and they won't make their stuff compatible with anyone else's unless the government forces them to. Maybe if we're lucky there will be a decade or two of highly public bitching (see also: the Tesla charging connector) until someone eventually capitulates.

Different battery chemistries have different charging requirements. So you'd have to have more complex charger/battery interaction requirements. Not insurmountable but another layer of standardization

Yup, I've been thinking along those lines as well. I can't believe that every manufacturer is doing their own standards again...