I think you might be focusing on the wrong site, Slin. If your intention is to make a better platform than tf2c/stadium/faceit then mixchamp should be exactly that. It would even do what you want it to already: balance the teams without captains.
I don't think it 'starting immediately' is an issue. Sure it's not perfect but 12 people shouldn't have to wait for another 12 people to add up just to have a bigger pool to balance with. That's impractical. I'd rather play 2 games with teams that might be less balanced than 1 game in that same timespan because there was no 4th medic for half an hour.
The only reason I haven't switched to mixchamp yet is because it's not alive when I want to play. This can be solved by promoting mixchamp more. The new map cups helped with that but it needs even more. Organize viewer mixes with the site when you stream, promote it in your videos, ask other streamers to do the same.
The newbie mixes could promote it as well. There's plenty of people there who'd be down to play more without having to rely on tf2center and aren't confident enough to add up to pugchamp
Mixchamp might not be 100% what you're looking for but it's still the best thing you can use by far to achieve it