There isn't any information on toweling because we try to avoid it. The only time it's recommended is if your bird is injured and needs to have an exam done. There's really no proper way to do it, as long as the head is sticking out of the towel they're usually fine. Other than that, we don't recommend doing it if it can be helped, it really tends to hurt any trust or training that you've began.
Hi James - my vet will towel Bennie when she clips his nails. She is very gentle with him and just does it for the nail procedure and exam. She's funny, too...after she's done and Bennie is on my hand again, she'll scold the towel and toss it down on the table. She doesn't say it loud enough to scare Bennie, and I think it's cute.
I have toweled Bennie a couple of times when I have clipped his nails myself, but I don't like to do it. Holding a struggling bird with one hand while trying to clip his nails without hurting him is a tough job! Stressful to both of us; I prefer to let the vet do it.
But I agree that is is something that should be done only when absolutely necessary. Cockatiels would not enjoy being cuddled in a towel like a puppy or kitten might.
I will say that the avian vet managed to towel Simon without there being much of a fuss. She referred to it as making a "birdie burrito". When I tried, it didn't go so well. Fortunately, he liked the taste of the medicine I was giving him and I got him to start taking it without wrapping him up. I wish I had the vet's skill, however, because sometimes it is necessary.