The two-party system is not a consequence of the lack of access to money. It is a consequence of the first-past-the-post system, which will always trend towards two parties (as a matter of game theory). See https://en.wikipedia.org/wiki/Duverger%27s_law.
The problem of primaries selecting for extremists could be partially alleviated with ranked-choice voting, so an extreme candidate with a significant minority of supporters can't win against a divided field of moderates. This is how Trump won the 2016 Republican nomination and how Sanders almost won the 2020 Democratic nomination before the establishment candidates dropped out and endorsed Biden. Or, we could try open primaries so independents can give input on the desired candidate.