It's been a while since I've used Teams+Sharepoint, but I vaguely remember being able to have a Sharepoint workspace per team channel, and then being able to mount those workspaces as a shared/network drive. Stuff that might end up in the workspace is like... shared/uploaded files or something?

This is correct. Every channel is backed by a Sharepoint folder (maybe this is configurable?) that contains stuff like uploaded images, videos, recordings and transcripts of meetings, etc.