Could have leaked out of the shrader valve!  Check it for leaks, put some spit over it and see if it makes a bubble.

This.
That’s always the first thing I check.

Not sure if this would cause your symptoms but I frequently have to clean my pressure switch due to sediment/calcium build up. I notice my pressure drops off and the pump cycles more frequently due to the switch incorrectly reading the pressure.

Definitely not this.  The pump is cycling as it should be.  Bad points or connections ain't gonna affect when the switch cycles.  
I'd guess he's hearing the pipe vibration when the pump runs.  
Random pump run cycles could be caused by a water softener regen cycle.  
Check the air valve under water pressure.  Turn off water and drain lines and check again.  If it ain't spewing water, add air and forget about it.  If it's spewing water, add air and plan to replace the tank soon.
If the pump cycles with all water turned off beyond the pressure tank, you probably have a slow leak.  I'd suspect a bad check valve down in the well or a cracked pipe inside the well.

With the exception of erratic pump cycling, my system works very similar to what OP is describing, low cut-in pressure, cycles every time we flush toilet.  I'll check air pressure and add more air when when it becomes more obnoxious.  
I'm pretty certain we have constricted screens at most fixtures and probably weak pressure switch response caused by dirt and buildup.  Have bypassed our softener because it failed after just a few years.
Our water comes from a developed spring, not a well.

I have a 100' submersible about 30 yards from the house.  If TV is off, I can hear it run.

Water logged

Water logged will reach a balance. The diaphragm is damaged, so the whole tank holds water. You won't see a pressure drop repeatedly.

Amen.

He said it was in the ground, so yes I assumed a submersible. Maybe I misunderstood, a jet pump would be audible.

OP, you did it correctly.

The fact that your system didn't drop pressure overnight seems to show that you do not have a leak.

You depressurized the system and set your tank at 28psi. This made things better so now monitor it to see if it slowly reverts to its old ways.

When bladders go bad the air will slowly dissolve into water and disappear.

Pressure tanks and switches have a lifespan. Eventually they fail.

Manufacturers recommend draining the system, checking the tank pressure and correcting it yearly.

It is HARD on the pump if the tank is bad. A pump can continuously run and be fine but rapid cycling will kill them.

Tanks aren’t that expensive. Just replace it. Also, don’t oversize it. Flexcon makes a good tank with their Challenger line.

Challenger

This.

Only possible issue I could see is if you well doesn't have the capacity to run long enough to fill up a big tank. In reality it should extend the life of the pump as well, less cycles.