Does "select a president [senator, etc] uniformly at random from the voting population" count as a voting system? I feel like that's about the only way to fix things at this point.
(I'm partial to approval voting (vote "yes" or "no" for each cantidate separately and if the winner doesn't have at least 50% "yes", the office is vacant until the next election), but I suspect the party line would immediately become "you're a traitor if you approve anyone but the Party's candidate".)
Also, yes, approval voting is a great approach and we should absolutely use it. We'd still get people who vote party-line, and it'd take a while for people to truly internalize that they can safely vote for other candidates they like, but it'd be a massive improvement.
(I'm partial to approval voting (vote "yes" or "no" for each cantidate separately and if the winner doesn't have at least 50% "yes", the office is vacant until the next election), but I suspect the party line would immediately become "you're a traitor if you approve anyone but the Party's candidate".)