Currently there's an input limit because of this "buffer" (how full the waiting room is allowed to be), when I got my first dose 6 months ago there was a long line outside of the vaccination center (not in the UK), and people showing up at 5:20 for a 5:30 appointment was told to wait (the first limitation was probably only having x bookable appointments per time slot). The healthcare workers were going as fast as they can though, so I guess that's the real bottleneck.
I guess without the waiting area they can add more vaccination booths, but geez, that sounds stressful for the workers, imagine having a person every ~2 minutes for extended periods of time.
Which makes me think I should move my booster appointment to the morning...