In order to make it mathematically possible for a third party to compete, we need to switch to something with more nuance than single vote, first-past-the-post winner-take-all elections. Ranked Choice Voting has some momentum right now, and AFAICT is no worse than any of the other options (they all fail in certain edge cases, I believe; it's just a matter of which ones).
And yet most actual Congressmen have a high approval rating within their district. Incumbents have an extremely high return rate.
The problem is never your Congressperson. It's always because Congress is filled with other districts' Congresspeople.
I don't think you'll fix that by un-gerrymandering. If anything, I bet you'll get even higher approval ratings for the incumbents, since you'll have fewer "cracked" districts (boundaries drawn to make a group a minority in two districts instead of the majority in one).
Ending gerrymandering might get a Congress that better reflects what people want. But mostly, what people want is for "the other guys" (whoever is not in your party) to win.
People might appreciate having had the chance to express their first choice, but when they're forced to settle for their second, third... hundredth choice, I'm not sure they'll be any happier.
There are ways to do away with the single-winner system, such as party lists. They, too, have drawbacks, but they'd at least be different drawbacks.
Even just allowing people to provide more than one vote means that people can support third-party candidates without that vote effectively robbing their preferred major-party candidate of a vote. (eg, if you're broadly left-wing, and like the Green Party, you can rank their candidate first, then the Democratic candidate second—and then if the Green Party candidate doesn't win, your vote counts for the Democrat) That's a big, big change.
I don’t think it does [1].
[1] https://www.law.cornell.edu/constitution/articlei#section4