In my experience, sick babies do not want to sleep in their crib. Nor in the car. They would be frightened and uncomfortable and want to be held.

Tylenol is not for treating the illness. Maybe the parents you're referring to don't know what it does. It is meant for decreasing the intensity of the symptoms. One of those symptoms could be a fever and potentially the body aches that come with that. But then you have to define the word "fever".

Giving the baby Tylenol as you're describing is probably not a good idea. Using it as the tool it is meant to be I would say is.