You have to do a mix for live recordings. What you hear out front and what would be recorded a very different, the vocals are normally WAAAAY up in the mix as is, quite often, the kick drum.
I'll get spoddy for a minute . . .
The way that I would have to do a live recording is to take a pre-mix direct line out of each channel (providing that is that the live desk at the venue has them) into my
VS2480 portable recording desk. That means that each track from the desk will record to it's own track which I can mix and piss about with later on. It's important to have a pre-mix direct out as I want the most untreated sound I can get, no FX, no EQ and most importantly for vocals no jacked up gain, I'll be looking to do all that sort of thing later. One of the biggest issues that I have come across doing live recordings in a small venue is that the vocal tracks will often be
really distorted, that's why I really need to avoid any gain being put on before it hits my recording desk.
A mixing a live recording is pretty straight forward so it will normally take a few hours work (although it's better to spend a bit longer on things if you can). I would send the tracks off to be mastered as I loath mastering and it's always worth getting a pro mastering job, it can make or break a recording. After that it would be ready to go.
So, to answer your question I don't think that I would be able to get a recording of a show out the next day as I don't have the time, skill or equipment to turn a recording around that quickly.
That said, I like your idea and there's no reason why I couldn't offer people a live recording a week or so after the event or maybe even at the end of a tour. It's something worth bearing in mind.