I think the rubbish teams tend to play a few youngsters and fringe players when they've no chance of the play-offs. So from their perspective, they want to play well to impress or they know they might be out of work the next year. Players are unlikely to care about helping the team get a good draft pick to increase their chances of being replaced.
I'd be up for introducing relegation though. Expand to London, LA, Toronto and San Antonio. Have a top league of 24 teams, split into four divisions of six teams, AFC East/West/NFC East/West. Lower division of twelve teams split into League 2 East and League 2 West.
Promotion and relegation to include end of season playoffs played on Thursday, Friday and Monday nights and in the bye week before the Superbowl, cancelling the tedious Probowl.