I live in a house that is about 15 years old. The submersible pump controller chatters in the basement when the taps are opened. I have checked the tank and re-pressurized it several times over the last 2 years. Some pressure but not much remains in the tank after several weeks. I can get water to come out of the air valve if I let enough pressure off of the tank. My analysis is that the tank has a leak in the bladder and once the air slowly leaks off and it replaced with water it does not have sufficient backup reserve so the pump must constantly cycle. The only thing to do is to change the pressure tank.
Answer:
By the sound of it, replacing the bladder tank is probably your least expensive route to go. I'm also wondering about the foot valve and from there, my fears include a lack of water supply to it.