Notes
Most recent cameras should be adequate, but I recommend your idea of separate audio recording.

Myself and 3 friends started a similar project a couple of weeks ago, but, because we meet infrequently, it has taken until now to refine the process.

First attempt; I set up my Fuji X-T3, pressed the button, then we played a few songs. When I looked at the video later I discovered that the camera has a 30 minute maximum record time and that the room reverb, while not apparent at the time, becomes very noticeable when recorded. (to enable distancing we were in a badminton court)

Second attempt; We moved to a smaller room and recorded in sections of less than 30mins. The reverb was less but still intrusive and the built in camera mics had difficulty with some frequencies, emphasizing highs and poor bass.

Since I am an amateur stills photographer I know little about video, so I did some googling and found that if you include a "clapper board" type reference at the start, you can record audio separately, in line it up with the camera audio track, then delete the camera audio.

Third attempt: While filming,Run all audio through a mixing desk and monitors, pan vocals and instruments hard L&R and record via stereo out to a Tascam DP-008. We had no effects added on the mixer as I found that reverb is sent in stereo, so instruments bled into the vocals and vice versa. I added slight reverb in DAW at mixdown.

This is the stage we are at as of yesterday. The audio is 100% better, and this allows a degree of control between vocals and instruments before final mixdown. Now that the technicalities have been sorted, hopefully we ca go for a "take" at our next meeting.

Hope this is useful,

Vintage