“if you operate an aircraft in this country, the FAA has to have some indication that you know what you’re doing. Particularly if its a commercial aircraft.”
No.
The aircraft and the pilots are certificated by the country of origin. The maintenance records for the aircraft, the pilot records, none of that is under the authority of the FAA. The FAA does do an initial audit and a recurring audit to verify compliance with safety standards to allow foreign air carriers to enter the United States. But pilots come and go during these periods, so, no, the FAA is not up-to-date with training records, deficiencies, etc.
So you don’t have to have a type certificate in the a/c your flying if it’s a commercial airliner? I’d be really surprised it that were the case. The FAA has to have flight standards for crews operating a/c coming into this country.