Good point about the open strings though. Less of an issue on instruments from other cultures like the kamancheh, where you would rarely play an open string, but certainly true on multi-strings where the open string situation is "oft-used".
https://en.wikipedia.org/wiki/Syntonic_comma#Comma_pump
Also, my understanding is that a big part of the magic of barbershop quartet singing is ringing (https://en.wikipedia.org/wiki/Barbershop_music#Ringing_chord...). Those chords that sound so pure that it gives you goosebumps. That requires singing in just intonation.
The interesting thing about the Barbershop 7th is that it doesn't function like a dominant 7th chord, which feels unstable and pulls towards the relative tonic chord (e.g. G7 -> C). The Barbershop 7th chord has a slightly flattened 7th to line it up with the overtones of the root note, blending it in and making it stable.
You might say it's a microtonal effect in that you hypothetically could notate and sing a C7 differently depending on its function in the piece. I don't know whether this is done in practice, though.
Does it really? Listeners are pretty forgiving, so I think in practice, you will just have drift away from concert pitch, and you maybe just have to be a bit careful because the open strings will jolt you back to concert pitch.
But in a capella vocal singing, there's no such forcing function, so you just need the ensemble to be good at locking in with each other, especially on long chords where you really want the audience to feel the consonance or dissonance. Which isn't simple, but it's also not rocket science.
It would be very hard to perform exact pitches within a couple cents of exact frequencies, if that were necessary. I just think theorists vastly overstate how much listeners care about tuning in most cases. I made this mistake myself. For research, I was interested in creating a sythesizer that allowed composers to switch between tuning schemes mid piece. And I could barely hear the difference, and much to my disappointment, it wasn't very compositionally useful.