Are Ground Crew Airline or Airport Personnel?
Ground crew are typically employed by a handling agency, which is a company dedicated to performing ground handling of aircraft (exactly the things you describe).

Some airports have several different handling agencies. Have you ever noticed, when waiting for your luggage to arrive at an airport, that there are several different help desks to choose from, depending on your airline? Those are the different handling agencies at that airport.

At a given airport, a certain airline will typically have a contract with just one handling agency. Sometimes, airlines have their own handling agencies. The exact structure can vary.

Smaller airports often do not have third party handling agencies, but instead offer handling services directly to the airlines. In such cases, the ground crew is employed directly by the airport, and could also have other tasks, for example as airport fire fighters.