What crowd? Based on a quick scan of my inbox, omitting a user agent is fairly common. In fact, it seems more common to have no User-Agent header than to have any User-Agent header.

I also expect that giving the user control over the textfield is even worse for privacy. If the user has multiple accounts, they will all use the same wack-tastic field they entered, and they will be the only ones with that User-Agent as well.

It seems best to keep this simple, whichever way we go with respect to the double-negative thing.