I have no idea what's involved, cost or otherwise, with setting up servers, but it would be an incredible first step if someone could set up a handful of 12- and 18-person public servers using competitive rulesets/unlocks, nospread, nocrits, ALLTALK OFF (!!!!), etc.
I am not a competitive tf2er. I watch streams, watch casts, try and follow the scene a bit but I'm nowhere good enough to play with just about everyone on this site. I know about lobbies, but the process of getting one going is burdensome and it's shitty when you're terrible compared to the rest of your team. I couldn't imagine a pub medic trying out a 6v6 lobby and how terrible of an experience that would be. A pub medic playing a pub frequented by good players, though, would be less daunting. He or she wouldn't be waiting ~30 minutes for the match to start and everyone to get in the server, at least.
'Competitive pubs' would also allow for players like me to be exposed to competitive maps that I see on streams. I think I could count on one hand how often I've played Snakewater and Gullywash. I've never even played Process. It was a bit disheartening to hear everyone at i46 shit on gravelpit because it's practically the only comp. map that I actually know and have played.
Again, no idea on the cost of these types of servers, but I would sure try and frequent them.