Given that an NFL player becomes an instant millionaire if they get a contract they quite simply don't care which side they play for or where they live.
Just to be clear, the NFL draft is actually a lot more complicated than people tend to realise. When a team drafts a player all they get is exclusive negotiating rights. A player can refuse to play for a team and sit out a season to re-enter the following years's draft. Obviously that is not in the best interests of a player or a team which would have wasted its pick, so instead the player can be traded to another team. Although most NFL players will have already spent 4 years away from home at university so they will have already got past the idea of playing for a local team. They have an attitude more like a rugby player offered a big money NRL contract, it does not matter which team it is you go wherever they want and make a lot of money doing something you love.
Also, other than the top round picks, players and not guaranteed to become millionaires in the NFL. Within the salary cap there is a rookie pool which specifies a minimum and maximum amount that can be paid on rookie contracts. This will often mean players from outside the first and maybe second round will not earn a million dollars for their first season.
And NFL contracts are not guaranteed. If a player signs for $50m over 4 years, it will usually be backloaded, so each successive season earns more, but they can be dropped at any point so if they have a bad season they could end up only earning $5m of that. This is why there are also upfront signing bonuses to offset the lack of security. While drafted players usually receive signing bonuses too, these are included within the rookie pool so are limited. They only make the big money if they become a success, which is not guaranteed as the step between college and professional American football is a big one.