It won't be the birth of a third party, it'll be the year the House decides who gets to be president. That's what happens when no one gets enough electoral votes.
Not with first past the post, it's not. Nor the electoral college. Trump or Biden WILL be president in 2024 so long as they are alive, regardless of who people vote for. Literally. It could be an even split between ten candidates and it will be either Trump or Biden.
That they will, but that's why it's so important to know where to effect change. With first past the post and ironically, after some progressive laws being passed, it's actually really important to go for one party or the other. At least in the states where the electors are in theory bound to vote with the popular vote.
You might say that sounds like a good reason to go third party, but it's still first past the post. If no one wins, then some selection of politicians pick, and they won't pick third party anyways.
The president is a shit show. If you want to engage in effective politics, it's way more about local and off-year engagement. Maybe if we get rid of first past the post or get other tallying methods in, we might have a chance at big seat change, but until then... They're sadly legally entrenched very well.