OP's situation is almost exactly the same as one of my bands, with the exception that BL our is a professional musician. The band formed through BL's weekly local open mic night, where the rest of us became a loosely formed house band. Of course, jumping up with other musicians to have a go at stuff with them became the norm, so jamming along to tunes wasn't something we weren't worried about.
Side note: Personally, I have only a rudimentary grasp of music theory but due to being an ex-DJ, I'm familiar with a lot of songs, so if I can hear it in my head, I can generally pick up the main bits pretty quickly.
We found cohesion as a four-piece playing dad rock crowd pleasers and started playing out under the name Doomsville (loads of videos on TB). One of the things we agreed on was no rehearsals. If BL had an idea for a new song, he'd text it to us with a key and we'd try it out next gig. Not every song worked right away and if it wasn't better by the third performance, it got dropped.
Even though, BL drop ones on us on the night. Like at the last gig where someone asked for "Venus" in the break, which he had in his song book, so he comes back and asks if we fancy it. I've figured out the bass riff before he's finished explaining it to the drummer and then off we go. And it actually worked. Audience loved it because how many rock bands drop (what to them is) a Bananarama song?
It's something the band has become well known for in the area. The audience know when we start a new song that it could go badly wrong (BL at this point will tend to build up introducing it around just how badly it could go), so cock ups actually end up becoming part of the entertainment. And then there's the times we deliberately play it wrong to try and put the rest off for a laugh.

That said, we do land the majority of tunes and have developed a pretty slick set, so it can work without rehearsals for cover bands.
Originals bands are a different story, mind.