I have some experience with deep wells, I've replaced both my deep (280') pump and my pressure tank twice due to the horrible hard water in my area. Iron.
While the tanks have never been taken apart, their a sealed unit, I believe the bladder separating air from liquid is the usual failure point. Mine rusted out, but could be because of the location next to my salt fueled water softener, IDK.
Never tried it, but if you remove the Schrader valve on the tank and water vapor comes out, I'd think that would mean the bladder is comprised.
I don't remember what pressure the tank should being running, but 5lbs is definitely not enough.
They are available at any big box or plumbing store and are relatively easy to replace.
But with all things plumbing, you touch it here and 10' away it leaks

I tested the valve, it has a very, very slight leak even after tightening it. Any loss of pressure from the valve at this point is not significant enough to gauge off a tire gauge.
I did pump up the tank pressure (with a bicycle pump) to almost 25 PSI and test to see if water came out but there was nothing, just what I’d consider slight common condensation.
My pump sits atop the (standing vertically) pressure tank so changing the tank will require pump and plumbing breakdown and reassembly.
I’m considering using my city water to try and fill the bladder by attaching to gate valve at the drain and pumping water in to see what happens.
One thing I’m not is good with plumbing so any replacement will need the pros.