You don't understand the bowl system. The 'integrity of the bowl system' isn't that every bowl matters to every other team, because it somehow has championship implications. The bowl system is important because it allows students (both fans and athletes) to get fired up about an extra game they get to play on a national level. The bowls (and college football in general) is about your team, your school, and your spirit.
Tennessee is not recruiting well just because they got 18 dudes