I'm not sure where you get the idea that American football has meaningful links to association football (American soccer). It shares a name, sure, but otherwise it is much closer to Rugby (Union/League) Football - both in its rules and in its historical roots.
However, I suspect the issue at hand is primarily that it's hard to define what would be called a "hat trick". Quarterbacks frequently throw three touchdowns in a game; so that's out due to its frequency. Hat tricks, in most sports, are difficult to achieve; for example, only 56 hat tricks have been achieved in Rugby World Cup history. In comparison, 179 QBs have thrown for 5 touchdowns or more in a single game, and over three thousand have thrown three.
As for other feats - it's hard to say why a phrase didn't catch on, but I suspect because there are simply too many things going on in one game. I've certainly heard the phrase use in its common meaning - "three of anything happening" - to refer to sacks and interceptions; it's probably been used in other circumstances as well. This is similar to other American sports; baseball often uses it to mean three strikeouts in one game, which is pretty silly (as that's not a good stat, and it's a very common one) and to mean three home runs in one game (which at least makes more sense), but even three steals in one game might be called a hat trick.
But in American football, what else are you going to call a hat trick other than passing touchdowns? Receiving touchdowns, rushing touchdowns, picks, sacks, made FGs... the number of things that it could be attached to is too large, likely, for it to have been attached to just one thing. The sports that it has become commonplace in are primarily sports with one major action - goal scoring - that is fairly infrequent. Cricket is to some extent the exception - and that's because it is the origin of the term, likely.