Remain conscious gives you a small chance of avoiding being slept. This won't matter to a vet because:

They won't let their opponent have the opportunity to cast it in the first place
If unable to do that, they will take precautions to mitigate being slept
Will have prepared sufficiently to have a good chance of surviving being slept

All the edge does is make a % of situations where you might have died to sleep into ones where you can survived - the vet's advantage is knowing how to avoid those situations in the first place.

So this edge being in the game generally makes things easier for those who do get into such situations.