ArxI think we would need to come up with some sort of 'hybrid' competitive game that still keeps the speed and skill of the existing competitive TF2, but gives public players something they can relate to in the game they play.
Replace "some sort of 'hybrid' competitive game" with "some sort of 'hybrid' casual game" and I would 100% agree. Leave 6's as it is, it's a great game mode. Bring in this "hybrid" game mode, not in hopes that the competitive players will cherish it and grow it into something great, instead create something completely different that allows for competitive situations within pubs.
Badwater is the most loved pub map because it creates 2v2, 3v3, and 4v4 situations(i.e. very good at spreading the battle out). Look at 6s. We have flanks and we have chokes. Each flank and choke holds 2-4 players each (depending on how teams hold). 2v2 to 4v4 is where Team Fortess 2 thrives the most. If you create pub maps (made for 20+ players) that naturally get players to fight in small team battles, you will have created those good "hybrid" maps.