That's certainly something worth checking. If the switch is old as hell, the diaphragm could be leaking and many time the pressure tube gets gobbed up with crud. Normally a 1/4" pipe so it doesn't take much. Usually there's a tell-tale and you'd see water dribbling out of it.
I would agree of zeroing in on the switch if it was ensured the water side was full and primed. It's still a possibility, but the point is, the switch is simply pressure activated. Does not always correlate to the mechanical pressure gage reading. If the switch "sees" 40 psi or below, it closes to energize the pump. If a pressure slug pushes it past to where it "sees" 60 psi, however short or long of time it is on, it will de-energize. The pressure fluctuations that I believe are from loss of prime is causing fake pressure spikes, touching 60+ psi then dropping back below 40 again. So it's going to cycle like crazy.
Troubleshooting from a distance and limited info is difficult.
Troubleshooting from 3 feet away is a PITA too for the uninitiated (me... or I guess I should say barely initiated. I lived 20 years on a well growing up and out house never had a problem aside from power outages.) It wouldn't surprise me if all of you are right and multiple things are wrong.
Unfortunately, when you weren't there when the problem first started, and, the guys who caused the problem aren't telling you exactly what happened and how because they're afraid of the financial exposure... that doesn't help things.
Being a shallow well could they have run it dry? Maybe. But they deny using any water.
So if we believe they didn't use water, why were multiple breakers turned on to activate it? Don't know, and they're claiming it was some sort of "honest mistake."
SOMEONE opened the well lid because they broke the concrete edges. But they don't admit to having fooled with anything inside.
I tried taking the easy route - calling the company that installed the last pump, and other in the area, and none of them repair this system anymore. It's a you're on your own or upgrade for a few grand situation. So, now it gbodywelltech101.
Fwiw, I videoed the noise/surging issues as the real world problem with functionality is there, but, I guess maybe doesn't stop there. As far as total system behavior, I watched it for a while, all PSI readings per the pressure gauge, you can take that for what it's worth.
When pressure falls to around 10-15psi on gauge, pump activates and pressurizes to about 35-40psi smoothly, before surging that was recorded happens. Surges have needle bouncing between 35-40 and 50ish psi back and forth, with the cycle time between surges getting very short before it shuts off. At shutoff pump is about 40psi, and slowly begins a leak down that, may/may not be driven by a now drippy hose bibb nobody knows anything about using. It will fall, without activation, back towards that 15ish psi or a bit below before the cycle repeats.