Most birds have a very poor sense of smell, so they are unable to notice human scent on baby birds (even a skunk’s spray doesn’t seem to bother many types of birds). In most cases, even if the nest is destroyed by wind or other means, you could create a new one and put all the nestlings back in it and the parents wouldn’t care that their baby birds were in a different nest when they come back; so long as the new nest near where the old one was so they can find it.
Hoopoe
Ducks and Hoopoe will often poop on their own eggs with particularly smelly discharge to discourage predators from eating them.