The loading time for a new character is only 2 seconds (at least on PC), so I'm thinking character info on the loading screen isn't necessary for how short it is. That said, it is silly how in SC4 and SC5 you would get the next opponent immediately rather than needing loading (and in SC4 you could also instantly tag between 2 characters), but in SC6 you have to load for every single character. I imagine this is more to do with the development team not wanting to do spend time adding support for multiple characters in memory than technical limitations, though.
I would be really happy if they ever added team battle mode to SC6. I got a small taste of how it could be by doing Hilde's Soul Chronicle in co-op using a mod forcing the AI to be human-controlled (skip to 30:17 if it doesn't already start there):
One downside to a team battle mode is that it makes some balance a bit weird. For instance, I'm pretty sure Project Soul designed Amy around normal first-to-three-wins matches. But team battle mode isn't a super-serious mode anyway, so I think that would be fine.