The fact that the pump can deliver 60 psi suggests to me that is is probably NOT the pump.
Here is something you can try even if the tank has a failed bladder.
1. Turn off the circuit breaker so the pump won't run.
2. Open a drain valve near the tank. You may want to run a hose outside.
3. At the air valve in the top of the tank (if it's a bladder tank) that looks like a valve stem on a tire (it may be under a plastic cap that you must pop off), add air until the pressure is about 35 to 38 psi (based on your description that the switch starts the pump at 40 psi). You may need an air compressor. It is possible but a lot of work with a manual pump.
4. Close the drain valve.
5. Turn on the circuit breaker and lift the lever on the switch as you described in your original post.
The pump should start and the pressure gauge should almost immediately rise to the same as the pressure of the air that you added to the tank.
Then is should rise normally (not "jump") to the shutoff pressure of 60 psi. Measure the time in seconds from pump start to shutoff and write it down for future analysis of problems.
If it works as I have described, but the air pressure when tested again with no water in the tank drops to zero within a day, then you have a leak in the tank on the air side of the bladder. You will also see evidence of this if the time to pump from start to shutoff gets much shorter than when you measure it as described above.
If it works as described, but the air pressure when tested again with no water in the tank drops over a few days, then the bladder has probably failed.