The NFL team would be miserable and probably have soft-tissue injuries, and eventually would not be able to stop the soccer players from navigating around them like cones, assuming they ever could. Assuming they could learn the byzantine rulebook in a reasonable timeframe, the soccer players would be instantly broken into little pieces. No one would enjoy either contest, and we would learn nothing.

As for why they didn’t catch on, first I’m not so sure they didn, as tennis in particular has always had its place in the American culture, though its association with the “country club” class may have limited its ceiling. American soccer has its issues, and it is not pressuring the “traditional” American team sports, but attendances are healthy, sponsorships are good, and quality of play is decent, with a starting 11 being roughly comparable to the bottom half of the English second division. Roster rules would mean an MLS club would quickly get ground into dirt in that English second division, but matchday 1 might be pretty competitive. Taking your question more generously, though, competition from baseball, followed by organizational disarray, followed by competition from college gridiron football, followed by competition from professional gridiron football, with the “not invented here” syndrome left it seen as a sport for immigrants and then as a safe yet cheap option for suburban children.

Televised World Cups and Pele started to erode that some, but more organizational disarray left the country without a proper professional league from 1984 until 1995, and when it was restarted it was intentionally done in a manner to control costs and favor management, which it ironically was able to do because it could always argue the players could seek employment in other countries.