I’m a bit confused as to what your ultimate goal is…
You use the term “demo” here, which would suggest a recording with some corners cut just to produce a rough representation of your band…
But the way you’re describing the recording/editing/mixing process suggests you’re actually aiming at a full-blown top-notch production. I think you first need to clarify your expectations.
If just a high quality demo is what you want, I’m not sure that paying out for a high-dollar studio would be money well spent, especially if you’re not confident the band could get it together sufficiently to leave with tracks you are happy with.
However, it sounds to me like you don’t want to get involved in producing & editing the band’s performances, but wherever you decide to record, that seems like it will be a significant part of achieving a result you will be happy with… Either that, or replacing the sloppy performances entirely.
That said, reading your description of the band’s various abilities, I was expecting something worse than the sound of the recording you posted above. It’s certainly not a car crash, and the songwriting and arrangements are pretty good.
You’re right though, heavy direction of the performances from a producer and post-production editing are inevitable if you want to polish this to the standards people are used to hearing.
My suggestion would be to convince the drummer to record his drum performance with midi triggers and replace the drums entirely with samples from something like Superior Drummer. (An E-kit is a good alternative, but not all drummers are willing to play them).
That way, editing the drums is as easy as editing midi, and you don’t have to worry about getting a big $ room to get decent sounds. Once you get the edited drum tracks, they could form the basis overdubbing the rest of the parts.
You could do all this without ever having to set for in a studio - big $ or otherwise.