It's not really the Presidential system that restricts the US to a very strict two-party system, but the use of primaries. With a primary system, any new or unusual political position can still be incorporated into one of the existing political parties by contesting and winning a few primaries. Without one, party leaderships can exclude people from becoming candidates and those people will then gather into third parties. Imagine if Trump had had to apply to a "Candidate approval sub-committee of the Republican National Committee" before being allowed to run for the nomination in 2016. He'd have been rejected, and he (and his political movement) would have been outside of the Republican Party. Similarly for Sanders and the Democratic Party.
The USA had a presidential system in the period before WWII too, but had lots of third parties - Progressives, Populists and Socialists all won significant numbers of local representatives (far more than any post-WWII third party) and had their own caucuses in the House at times. Minnesota's Farmer-Labor Party was so successful, the Democratic Party had to merge with it. But primaries, with open filing rules, for most US House and Senate seats and most local elections were established through the 1920s-1950s and they brought all sorts of people who the party establishments hate into Congress and into state legislatures.
Obviously, the US would have two dominant parties as long as it keeps the FPTP electoral system. But there's a difference between a normal level of dominance like Labour/Conservative in the UK or Liberal/Conservative in Canada and the extreme level in the US, and while the Presidency is part of the problem, the core one is that third parties can't build benches because they can't win elections because the people who would fight and win elections for them are, instead, running as radical candidates in major party primaries.