Well, this is timely: Bruce Arena has resigned (not quit, though of course we don't know how strongly he was encouraged to leave, and how much was his own sense of responsibility).
In a case like this, I do think a new coach is probably right, simply because on a national team it's a lot harder to just get new players. Some countries that are more football mad, it's probably a lot easier since they've just got more depth of talent in the pool of eligible players, but I'm not aware of any stars or any hidden talents in US men's soccer (in contrast to women's soccer, where we've got plenty of players in the NWSL and around the world who have the quality to make the team, we may well have fewer than 11 really world class men, though that may just be my own biases speaking). So, if you accept that we can't make meaningful player changes (you don't have to) and you can't really change the 'ownership' (though I do think the organization isn't particularly good in many key ways and does need to make some changes) then all you can really do is change the coach.
Now, of course, the counter point is that, especially for a team like the US which (despite two decades of World Cup qualification) is really still in need of development, changing coaches affects players and the time being able to grow and develop. How is the team to train? How do we bring up our young players? Are players encouraged to go to and recruited from Europe, or are they encouraged to stay in the MLS? Of the 23 players called up for the most recent game, 15 were over 30, and only 5 were under 25, which hints that the next coach will have a lot of world to do to continue to attract and develop talent (now, a dozen or so other young players have played with the team in other games, a few in quite a few other games; why Arena picked the team he did, you'd have to ask him. Also for reference, all our opponents were under 30). So, one might hope that we can identify a good coach and give them time and let them endure some setbacks, so long as we retain confidence in them, and let them work their magic and shape the team with their vision.
But, how common are such coaches? Where do we find one? Go abroad again? Hire locally? I don't know.