I think this would be a terrible idea.
1) Tanking is NOT a big issue in the NFL. There's only a few examples of teams tanking an entire season in the past 20+ years (IE Colts for Luck or Browns for Garrett). There's been any number of janky final games of the season like our game vs Pats last year, but I don't see how this proposal would help that.
2) This would make tanking MORE blatant. Teams would be in a desperate scramble to make it to the minimum number of wins to keep their pick, and as soon as they reached it they would trade/bench everyone to ensure they don't win any additional games.
3) Any team that loses what would've been an early 1st round pick is going to be severely crippled going forward. Can you imagine being a fan of a team that can't make it to the minimum number of wins and then loses their 1st round pick? That team would lose all it's fans in a few years and fold...