Hi! PFH is correct and I have a little experience with this. First, congrats on getting an offer from a sport in Team USA that will actually pay you a salary. In my sport the trainers tend to get their expenses reimbursed, but they don't actually earn anything so they are all either independently wealthy/retired or they work a full time job in addition to being on the training staff.
My advice would be not to make the decision based upon the idea that you will get to see the world. The athletes, coaches, and training staff rack up a lot of miles, see a lot of airports, and get their passports stamped but they don't really get to see the places they visit. My husband just finished his 2nd Olympic run and was an alternate for both London in 2012 and Rio in 2016 but he didn't really get to see the world. He flew around the world multiple times, but he complains that he never got to experience it. He has been to Germany 4x, Turkey 2x, Brazil 3x, Canada 2x, Hungary 2x, Japan, China, Korea, England 2x, and a few more that I can't remember right now, but he says he doesn't feel like he actually visited those places. He just visited a bunch of gymnasiums around the world, and the big difference was that each one had different toilet facilities :D
That being said - if you can financially take the pay cut, having the work experience of being the head of sports med for one of the teams would be great for your resume in the future. One of our team's trainers has his own sports med clinic and I think part of its success is that he is constantly talking about the athletes he has worked with. The walls of the clinic are covered with signed posters and pictures with the big name athletes he has worked with through Team USA.