I actually think you can play exclusively with friends, although you’ll still all be in the public hub area (it’s minimal interaction, mostly shops). Otherwise, once you select to play a match, you’ll be loaded only with the friends you’ve invited and bots to fill up team comp (4 players at all times). As for Solo mode, yeah, no such thing.

I’d also argue for playing Public. I have about 800 hours in the game, doing exclusively Public runs. Gotta say, it really is fun most of the time. Sure, ran into some exceptions, but they’re rare. Had a single rage quit because one of the team members kept spam-marking enemies and backseat driving for the entire match, but that happened only once in several hundred so far. At worst, it’s someone who rushes ahead (and gets themselves killed, which you can safely ignore if you and the other 2 team members stick together), or people who’re not yet used to their gear/movement/enemy prio, but they at least participate and stick close, so you can always help if needed.

Edit: double-checked, and yep! There’s a Private option when selecting missions, which allows you to invite friends. Once that’s done, you can Merge the Strike Team (definitely have the option post-mission, dunno about pre-mission because I’ve never tried it) and it’ll make the team composition persistent across runs.