That's not it. It may be helpful for the Americans in this thread to look at the UK, which uses FPTP but has meaningfully successful third parties anyway. The lack of them in the USA isn't to do with FPTP but more fundamental.
The reality is that politics is essentially bipartisan. The left/right divide is the core of politics and the only one that actually matters in the absence of regional independence movements. Therefore, what we see is that third parties usually struggle to differentiate themselves to voters.
This can be seen most clearly with the Lib Dems who are one of the main third parties in the UK. They tend to be used as a protest vote and few would be able to articulate what they actually believe. Arguably they believe nothing at all, as they have in the past reversed their position on core manifesto promises, fought against implementing referendum results (despite having "democrats" in the name) and so on. What do the Lib Dems actually stand for? What is the big difference between them and Labour or the Tories? It's unclear.
This doesn't mean you can't have different third parties. You can. The SNP is a clear example of a third party that grew from nothing to take over Scottish politics. But it seems that regional independence is one of the only things that is big enough to cause this, as typically the national level parties will not want to support a regional independence movement (would reduce their own power and take up a lot of time), but voting is regional.
Even there though, the SNP is basically just a clone of Labour plus that one additional policy. If Scotland were to become independent, you'd rapidly see the two party system re-establish itself there along left-wing and really-left-wing lines, because this is the only axis that matters.
In Europe where FPTP often isn't being used, you do see more parties, but this doesn't result in less partisan or more harmonious politics. If anything it seems to be slightly the opposite. Germany for example has major problems with people publicly debating whether the AfD should be banned, there are physical attacks on its members, and although PR is supposed to encourage collaboration and compromise none of the other parties are willing to work with it despite it being the second most popular party or even in some parts the first most popular party. The AfD is not particularly extreme though, and its policies aren't so different to the Republicans in many ways. This problem crops up in other European countries as well, it's not unique to Germany.
So a third party wouldn't fix anything in the USA. Nor would PR or more exotic voting schemes. People might occasionally use them as a protest vote, as they do already today, but there's no fundamental third ideology for a third party to express, and anyway who needs a new party when the two big ones are so internally democratic? Easier to take over one of them than make a new one.