In many other democracies, where third parties are more common, you tend to get the following sequence:
- Elections happen.
- If some party has an outright majority, they take control, nice and easy.
- But sometimes nobody does. Any change in leadership is a simple majority vote in the chamber. So the parties start negotiating between each other to see if they can form a coalition. Smaller parties essentially promise to support bigger party X (and/or leader Y) as the leader of the chamber/government, and in exchange they get certain positions of power, and/or legislative promises and concessions, etc.
- As long as an agreement can be struck between a majority of voters in the chamber, the new leading party/member is then determined.
But the US system essentially does this in reverse. Politicians and political hopefuls seek out which group of people they're willing to work with and compromise with from the start. This will form their coalition, either the ruling one if they're in the majority or the opposition coalition if not. As any coalition system devolves into "Coalition that has the majority" versus "Everybody else", two pre-constructed coalitions naturally arise. These are our Democratic and Republican parties. They don't really resemble the parties in other democracies, like the UK and Italy, say. They are assemblages of diverse interests, where members may frequently have strongly divergent opinions on certain issues. If you listen to much of US politics, you may hear terms like "Tea Party Republicans" and "Fiscal Conservative Republicans", etc. come up, which reflect that the Republican party is really a coalition of several parties (the same holds for the Democrats; AOC has been known to say that in any other country she and Pelosi would not be in the same party). They are maintained as a cohesive entity in much the same ways as coalitions are elsewhere: you get a little bit more power and a stronger support network in exchange for compromises and support on certain legislation.
In short, we construct our coalitions first, and then hold the elections to figure out which coalition is going to be in the majority. As such, third parties are rather superfluous, and serve little purpose other than to make it clear to the two main parties that they may need to shift their goals and priorities a bit to achieve a majority. And if they do so, the third party ceases to have a reason to exist, as its core issues and supporters will have been integrated into one of the main two parties. A few are able to make principled stands and sustain a political career as "Independents" (or proper third parties, at the state and local levels), though they are the exceptions. They still end up picking a coalition to side with so that they can have the influence and power necessary to advocate for their constituents; they simply choose to not label themselves with that coalition's name.