Another possible reason would be the difficulty of mixing four channels on the fly. While they could, conceivably do the mix during a rehearsal, invariably, nothing gets played exactly the same way twice. Today's technology allows the process to be easier.
I have done it.
We were making CDs of weekly performances, not direct-to disc LPs, but I mixed all of them in RM surround sound on the fly.
I usually used the same pannings week to week for the same parts, so the setup was usually about the same. I just had to find extra pannings when they brought in extra parts (e.g. a dulcimer).
I usually used the same pannings to make an RM/stereo-compatible recording:
Center Front: Lead vocal, bass, drumkit
Left Front: Lead guitar
Right Front: Rhythm guitar
Left Side: Keyboard
Right Side: Percussion and extra instrument
Left Back: Left half of backup vocals
Right Back: Right half of backup vocals
Center Back: Reverb effects
Stereo channel strip: Surrfield mic hung 20 feet above audience
I used the encoding method I described earlier