Why does the pump cycle on and off very quickly all the time any faucet is open with a none bladder water tank?

I believe that your pressure tank is "waterlogged", meaning, there is very little if any air trapped at the top of the pressure tank. A simple solution is to shut off the supply to the tank. Then completely drain the tank. And, finally, refilling the tank which will trap a sufficient amount of air, which will compress under pressure, allowing your system to have a larger operating pressure range. Good Luck.