Yeah Stone Roses came to mind for me too, in addition to the ones I listed. Also Ultra Vivid Scene, Primal Scream, Love and Rockets/Tones on Tail, and others.
For New Wave, again you will probably want a dedicated synth player, but lots of layered synths is just a normal thing a synth player does. Very few play all the parts live.
We’ve been around this loop, in fact continue to go around this loop, on a regular basis. Where we have got to is that we want to be a pub / bar band and play covers from the 60s / 70s / 80s / 90s and 00s that work with 1 x drum, 1 x bass, 1 x guitar and a vocalist. As a result, the songs tend to be more indie / rock(ish) orientated than, say, synth orientated. They have to be well known and have to be songs that people will enjoy and ideally move to. Oh, and a key factor in song selection is can each of us actually play our part? There is no point picking a song that the vocalist can’t sing, even if the we shift the key, or the guitarist, the drummer or I can’t play. There have been a number of songs suggested that I have simply vetoed on the basis the fact that they are too hard…
Our list waxes and wanes. Suggestions are made, then some fall off the list and are replaced by others. Our ambition is to get to a solid set list of c. 35 to 40 songs. We are currently at 23, with a further 7 being worked on (as in we can play it, but need to tighten up a bit). We then need another 5 to 10 songs on top of that. The guitarist and I came up with a list of 14 last night. The drummer vetoed 2 and the vocalist is yet to really comment…
As an aside, both the vocalist and I seem to watch quite a lot of YouTube content regarding how to “be” in a band and how to create a set list. Something I watched a week or so ago made an interesting point. Take song like Venus, Mustang Sally or Mr Brightside. A lot of bands play these songs, but … and here’s the kicker … the people in the audience a) will know the songs, b) might join in and c) are unlikely to be in pubs, listening to bar bands every night, so will not have heard Venus / Mustang / Brightside x times that week / month.
Anyway, we are playing an open mic tonight where we will test out 2 or 3 new songs - ones we haven’t played in public before - alongside a couple of old faithfuls 
Yeah. I think doing upbeat alt rock from the 60’s through 00’s will probably do us fine. I think the guitarist was more worried about us having to be more defined than that.
I’m mildly annoyed at the guitarist wanting us to narrow down the style we’re playing, when he’s the one most prone to launching into a random song out of left field, and then being the one most likely to shoot down a song - not because he can’t play it, but because he doesn’t like it.
Like, I suggested we should work on some Pretenders songs because they totally fit they are the new wave/post punk band we most naturally sound like, and he shot that down because he doesn’t like the Pretenders. Then he randomly launches into Smashing Pumpkins “Tonight”.
Another thing we have kind’a reconciled ourselves to is that as individuals, we will likely ‘like’ about 80% of the songs, but the important thing is that we play songs that punters will like. Tonight, one of the songs we are playing “All The Small Things”, it’s not a favourite of mine, its a song I tolerate, but I’ll play it as the other like it and it seems to go down well.
Our guitarist also randomly launches into songs at rehearsals. Sometimes, it’s ooh, yeah, we should add that to the list. Other times, it’s err, no! 
To be fair, he’s a great guitarist and knows an awful lot of songs!
Another way to make things interesting for you and the audience is to mash up songs or change them up to surprise the punters
We play Jolene at 170bpm! That’s a 1/16th note workout for me but it’s only for 2 1/2 minutes 
And these mashups.