It doesn't prevent it, it just makes it extremely rare. The reason is, essentially, Tragedy of the Commons: for each individual voter, the most strongest result from voting for a minor party candidate rather than the least-opposed major party candidate is to make it more likely that their most-opposed major party candidate will win over their least-opposed major party candidate.
> There are usually a few independents in either chamber of Congress
Since WWII, there's sometimes a few independents in Congress in total, but rarely more than 1-2 in either chamber, and often none in one or both chambers. Things were a little different earlier (in part because the two-party system in each state was sometimes not the same as the national two-party system; e.g., in Minnesota for a considerable period, the two competitive parties were the Farmer-Labor Party and the Republican Party.)
And even the very small number of independents in Congress overstates the non-party influence; very often the "independents" in Congress are either people who were elected as members of one party and then defected after being elected (sometimes being re-elected as independents once they have secured the advantages of incumbency), and at other times they are independents that one of the two major parties has chosen not to oppose in general elections. (Note that the two independents currently in the US Congress, both in the Senate, include one of each of those descriptions.)
However nowadays the two major parties certainly do cooperate in ensuring that there remain only two major parties, rather aggressively going even further than FPTP would otherwise lead to in keeping additional political parties weak. It's easier to get elected without a party affiliation at all than to be elected as a party other than D or R.
Whether it is "deliberately engineered" to do that or not, it has a fairly well-demonstrated effect of doing that compared to systems which provide more proportional results. Intent is pretty much irrelevant.
You are quite right about the key difference with the US having an elected executive vs the parliamentary democacy system of the UK, but I don't see how FPTP prevent the election of Congressional candidates from outside the two major parties. There are usually a few independents in either chamber of Congress, but I don't see any particular reason why another party couldn't make a showing if it were willing to put in the organizational leg-work.
In my (limited) experience American politicians and parties are excessively focused on elective office; being from Ireland myself I think long-term success stems from building a solid local constituency. So for example, I think it's a complete waste of time for the Green party to run candidates in the US presidential election, instead of working to capture a few seats in Congress or in state legislatures. I have been involved in a few such local campaigns, and what I've observed is that on failing to win a major election the candidates tend to exit politics instead.