Not even going to teams with losing conference records, teams with less than 20 losses receiving at large bids has certainly anything but rare in the 2000s. I picked 2005 and 2006 randomly.....in 2005, 4 teams with sub 20 win seasons were given at large bids. In 2006, 5 were.
Secondly, under your requirements it would be better for power conference teams to cut back on the number of conference games in order to allow teams to schedule enough cupcakes to get to 20 wins.