Hey everyone, OP here. Thanks so much for the replies! I've read all of them and learned a lot, but I don't think I can possibly reply to everyone, so I'll try to reply to them by giving an update on our session last night and other details. And, by the way, I did manage to work out something with the DM after posting this to take over DMing for the group soon! Yay!
Last night was a surprisingly different game than every other session. We wandered from a little port hamlet to the main city and unexpectedly there was some kind of revolutionary coup there and we had to decide if we wanted to join sides with the incumbent or the citizens. As far as story goes, this was the first we'd learned anything about the king character and it was a surprise that he had something to do with someone else we killed deep in the woods (a "boss" that went down in like 1.5 rounds with no additional baddies to fight). He had us play a kind of wargame that reminded me a lot of Stormcloaks vs. Imperials in Skyrim, with different parts of the city controlled by one faction and we had an abstract army that we would send one way or another. His rules for the fighting were very complex, so I shouldn't even bother explaining it all here, but it was highly non-standard combat that usually involved us killing about 5 or 6 trivially easy guards per sector with a time allowance of 2 rounds. Each of the guards were abstractions of some greater number of guards, but sometimes even if we won decisively we'd lose like half of our force in that sector because of his d20 rolls that had something to do with it. At one point he had to call time and chill us out because we all got really frustrated with rules he didn't explain biting us in the ass (e.g. no long rests between days or enemies just popping out of nowhere 10 feet in front of us). As far as enjoying playing our classes and using our character strengths, it felt rather pointless, but it had some interesting aspects like an easy-mode RTS game. My gnome wizard got to propose a (Napoleonic) military strategy that I think paid off pretty well for us. Next session will involve fighting the king and his elite guards, but for gameplay reasons he doesn't want to let our 100+ person army at the doors help us as we go inside to kill the person they're trying to kill. "Sure....."
Anyway, I had actually been working on my own elaborate Faerûn campaign setting since before I even joined this group. I've had 6 months, off and on, to work on it, think it through, and world-build. There were a few chances for me to host a one-shot for the group, but those all fell through for various reasons. Since it seems we'll be wrapping up this chapter of his campaign in a few weeks by sailing off the damn island, it was the perfect timing for me to propose running my campaign for a minimum of 4 sessions. I'm getting hints from him that he would be cool if my campaign lasted for as long as players had interest, instead of us returning back to his world after some number of sessions. I think he's getting excited about his own character he'll be playing. I'm in the process now of helping everyone figure out their race/class/backgrounds. I think the group is ready for big change and I'm very excited to be getting behind the DM screen! Perhaps I could have done this sooner, but I'm glad I haven't burned a bridge with the DM so far since I think he'd be a good player. Next week 2 of our 5 players are out of town, so I'm going to run a session 0 prequel session for the 3 folks that will cover how they meet, etc. It's a lot to come up with on one week's notice but I have plenty of free time at the moment. After that we'll try to finish the other DM's campaign, hopefully with a bit more urgency.
I think I agree with most of you all that a DM should definitely have a good grasp of the PHB before deciding to DM. I think he expected us to be okay with a very rules-light version of D&D because in his mind "rules are for fools", but he adjusted a bit to incorporate more D&D-esque mechanics when he realized we actually liked D&D. I'm taking a very different approach to my DMing: very classic Faerûn D&D setting, classic tropes of story involving a major conflict between elves and humans, classic monsters, well-thought-out and challenging combat scenarios during most sessions, full D&D RAW, etc.
A big takeaway for me from this 5 month experience seems to be learning what not to do and how important the crunchier parts of D&D are. This thread has been very helpful for me to feel a little vindicated in my higher expectations and these tips are great to help me transition into my own campaign for the group. DM flaws notwithstanding, we did have enough fun each session to justify spending an evening every week doing it. I think others players' expectations are artificially low because for most of them this is their first game too.