"The problem is that Congressional districts are gerrymandered as fuck, making it quite often impossible to have a genuine two-party contest within electoral politics. I mean, are you suggesting Team Pepsi run in Team Coke's Congressional primaries as moderates rather than contesting a general election they basically can't win?"Congressional districts are ultimately decided by the individual states, (theoretically) based upon Census data as I understand it. (A correction is welcome if I've misunderstood)
Given that, it seems like if you want to fix the US Congress, work must begin at the individual state level to ensure gerrymandering is eliminated. This would likely involve not only pushing for state representatives to support this but electing governors (in cases where the governor has a say in districting issues).
At first this seems like a "it's turtles all the way down" kind of problem but I think voter education campaigns (as opposed to the voter-insulting or voter-shaming that seems to be standard for the national level these days) would probably be the most effective way to address it. Regardless of party, I think most Americans are in favor of fairness and equality.
Trouble is, when a party wins (and thus is in the best position to fix the issue), this isn't seen as a problem worth addressing. That change, too, needs to come from within.