The recheck baggage personnel there are airport contractors. They do nothing more than scan the bag tags with a laser pistol gizmo, wait for a beep out of that same device, and place the bag on the conveyor (and sometimes, let the passenger to place the bag). They don't review boarding passes, they don't judge timing of future flights or eligibility for passing through to the TSA re-security upstairs or consult a computer terminal. So I suppose this all reverts back to your original airline checking bags through to the final destination. Beep, go on, no issues. No beep, I'm not sure what happens. I think they just refuse the bag and point you to the other airlines' recheck desks about 30 meters away.
I find the "TSA recheck" security system upstairs a little easier than standard security at other terminals. There is no pre-check though. I don't recall them checking boarding passes either. There's no official TSA document check like the regular lines. If there's anything, it's just a contractor looking for anything that looks like a boarding pass.