First sale doctrine has nothing to do with this. First sale doctrine applies to resales, not the initial sale.

Nope.

First sale doctrine explicitly limits the restriction that the seller of a copyrightable work can place on the buyer.

It is illegal for you to sell me a book and say "you can't let any of your friends read this copy of this book" because that violates the first sale doctrine.

I'm aware. But the fact that limits exist to copyright does not in any way show that this particular limit exists or is good.