What happens if you check in for a flight but don’t make it to the plane on time? Do they take your bags off or leave them there until you come back for them later?

The bags go to your Destination. Except for International flights. Flights doors close at 15 minutes before leaving the gate. Most Airlines stop you at the main counter at 30 mins because you can’t make the flight. At smaller airports a little less cutoff time is allowed. If an airline has your bag then that means you had time to make the gate. If there was any question about your bag then it would have been placed on the side and not loaded until the gate agent instructs the ground crew to do so. The plane has to leave on time. If you checked in on time and for whatever reason lost track of time and missed the plane, your bags will be waiting for you at baggage claim at your destination city.