The older ones I've seen haven't been any more or less reliable than newer ones, but they do tend to get renovated every other decade or so. The things that tend to wear out are the sensors that tell when the doors are activated or the location of the floor. The wiring that goes to the car tends to have problems because it moves all the time. Wires get old and crack and connections start to fail. In typical installations you have one small room on top of the elevator shaft one floor above the highest floor the elevator services. This houses the electrical service, the motor, the gearbox, and the control circuitry. We call this the elevator penthouse or machine room.
The earliest electric elevators didn't have control circuitry that told the elevator what floor it was on or exactly where the floor was. So they required an elevator operator who had a big rheostat (I think) type control that fed an AC motor(again I'm guessing) in the penthouse. I think an elevator from the 20's would be of this design. More modern controls were added as the elevators were renovated.
A typical renovation replaces all the sensors, wiring, and control circuits. This often requires a new panel inside the car. Often the motor, gearbox, and brake are replaced or overhauled along with the steel cables. The car itself is rarely replaced on an elevator, along with the rails inside the shaft. Pulleys and/or bearings are replaced or overhauled.
Most people who have their elevator renovated get an elevator company to do it. Most states require periodic inspections of public use elevators by certified elevator repair people. I don't know if these certifications are required to work on or renovate an elevator. If our elevators require anything more than a reset, we call the elevator folks with whom we have a contract.
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