It can be confusing when you first try to understand the American medical system: it's so different from what happens across the rest of the world. The important point to remember is that it is, despite the government actually paying for a good chunk of it and the insurance companies most of the rest, a free market medical system. This means that the various providers are all in competition with each other to try and get the business of tending to your medical needs.
So, unlike the UK for example, where if you need travel vaccinations you just go to the local general practitioner who deals with almost all of your other first line medical care, if you were in say San Francisco, you'd need to do a search for a travel clinic sf. This would give you those groups and physicians in San Francisco who specialise in travel vaccinations.
This is pretty much true of all of the US medical system. That specific tasks are handled by experts and specific clinics much more than is common in the health care system of other countries.
Yes, of course, they do have general physicians, you don't need to try and work out what problem you've got on your own before you can find the correct specialist. For example, a search for doctors manhattan would give you a listing of such general purpose physicians in that part of New York.
But as I say, the general way to look at the US medical system is that there's a huge network of different specialists, each chasing your custom. Which is one of the reasons why, as long as you can afford it or have the right insurance, the system is so good of course.