Another Civil War - that’s what it took to change the names on the doors the last time it happened. But it won’t.
What platform would a third-party candidate use that one or both of the majors wouldn’t co-opt? What popular demand for change would there be that the majors couldn’t recognize was there and claim as their own, with the subtext that they could make it happen?
That’s how the process has worked, historically, the ones that have been based on a more than simplistic appeals to “they’re both corrupt, both the same, we’re not them”, that is. The Populists and related regional groups found their platforms taken over by the Democrats, the race-baiters appealed to by George Wallace found a home with the Republicans. The budget-balancers Ross Perot appealed to found budget-balancing adopted by both parties, at least for awhile (it was always in GOP rhetoric but never acted upon by either party).
What looks like rigidity and staleness, when you first look at the two-party system, is actually resilience and adaptability - what the Dems and Reps stand for now has little to do with what they stood for 50 or 100 years ago, except that they have historically staked out somewhat different positions along the big-government/small-government axis, with a lot of overlap (less now than at other times, though).
With a multi-branch system, with the parties having little to no constitutional or even legal status (yes, really), it may be inevitable for a two-party system to spring into being, and in fact we’ve virtually always had one when there wasn’t a Civil War. People of similar mind will tend to support and vote for each other and each other’s proposals, and suddenly you have a party. People of different minds than that will be concerned that they’re gaining too much power and taking over the process with ideas that they don’t like, and they’ll band together and suddenly you have a second party. Anyone left is out of the game, and will associate with one of the parties in order to have any influence, recognizing that the party structure is now part of the political structure.
That works for candidates for executive offices, too, except that the factionalism occurs among a different political group, the electorate. If you want to gain the support of a majority, you have to appeal to the same things that, surprise, one of the parties appeals to.