I always stay with my birds when the vet has to do anything, and make a point of doing so. I like to know exactly what is going on. I sometimes help the vet with holding the bird or a wing etc, while it is aneasthetised. It's interesting to see how the breathing rate falls or rises within a few* seconds*, in relation to how much aneasthetic is being given. The dose/rate is varied using a dial on the gas supply line. For light aneasthetic (as when imping) the birds do come back round very quickly.
With some hawks, they don't even use aneasthetic, for imping sometimes; just put a hood on them, and in this 'darkness' the birds seem happy to be restrained without aneastetic while the vet does what is needed.