Why would it in a first past the post system?
Well, the UK and Canada both use FPTP and have semi-relevant third parties but they are also parliamentary. Similarly France has a similar-ish political system to the US but with 2 rounds elections and has third parties
I think it is the combination of both 1 round presidential elections + FPTP that makes third parties completely non-existent.
This+Electoral College
A third party is simply not viable without eliminating the Electoral College.
Unless...maybe you can keep it but you switch to a 2 round system and the two candidates with the most electors get to the 2nd round. But even then in practice i think it would be hard to convince people.
The First round could basically be multiple parties run and the two candidates who win most congressional districts advance
The Second round could be the election between the top 2 with the electoral college