> I don't see how FPTP prevent the election of Congressional candidates from outside the two major parties.
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.)