I think that the NFL takes gambling by players very seriously, because it truly could affect game outcomes, especially at the QB position., The QB touches the ball on almost every snap. If a QB were influenced by his own betting, or someone he owes money to from said gambling, it would not be hard for him to throw a game.
Look at the investigation/prosecution of the 2 Guardians pitchers. They just agreed to certain outcomes for certain pitches. I certainly don't want another such embarassment.
I think the biggest potential gambling problem the NFL faces is with the refs. It's way bigger with the refs.