Now I'm confused. TSA runs background checks on airport employees, right? ICE/USCIS doesn't, right?

Yes, you are absolutely correct! All new employment are required to being screeened all criminal background checks. if you are process to be cleared from FBI to be completely screening. That you are allowed go to work anytime soon and also, you will have go to security class to do take the tests. if you are passed the tests. You will be receives SIDA badge. You will have access into the ramp.