Do you know what ICAO is?
One of the key points of ICAO is that all memberstates have to accept that flights by other ICAO memberstate airlines can freely overfly their territory.
The only restriction they are allowed to make is to create corridors and no fly zones, but then these have to apply for all companies (to avoid protectionism).
To make sure this global system of free passage works with a miminum of safety, all ICAO memberstates have to make sure their local companies meet certain specific international standards.
If for instance Belize lives up to these international minimum requirements, then the FAA can not refuse carriers based in this South American country, or they'd violate ICAO priciples, which in turn could lead to ICAO revocing US carriers the right to overfly other countries....
BTW, Wolfpack, you apparently see foreign carriers overflying the USA as a potential safety risk, basically because they are not resorting under the FAA.
For most Asian and European carriers this is irrelevant since their countries of origine all have authorities similar to the FAA. For other carriers (from the 3rd world) you might be correct, but, have you ever thought about the fact that US carriers overflying a foreign country also involve a safety risk to that country?
I mean, Lockerbie for instance was an attack on US interests, but a lot of British people on the ground were also killed/injured in that attack.
You see, we all have to accept some kind of risk, or we simply couldn't fly anymore.
That's why the FAA has to accept foreign carriers in their airspace, just as other countries accept US planes.