Did the problem just start or has it been that way for some time?
As you found, the accumulator pressure should be set at or a couple of PSI below the pump's cut-in pressure. Check/adjust the accumulator pressure with the pump off and zero pressure in the water lines (faucets open, no water coming out). To clarify, you said it doesn't matter what the pressure is as far as pump operation...Did you check accumulator pressure with NO water pressure in the system? If you have the pump turned off/faucet open/system water pressure zero, can you charge the tank and will it hold pressure? If the bladder is ruptured, it will hold no pressure at all. Also, as noted, if the bladder is ruptured and there is water pressure present, you may get water out of the Schrader valve on the tank. BUT you might not because the tank may never actually fill with water even with a ruptured bladder because there is air in the upper part of the tank that simply gets compressed by the water. SO...it's possible to read some pressure on the tire gauge that is the result of the air in the tank being pressurized by the water.
FWIW, 40-60 PSI is pretty high pressure for the boat's system. I don't think that the OEM GM pump (if that's what you have) is designed to operate at that pressure level. This can contribute to the problem of rapid cycling. The OEM Hatteras recommendation is 20-30PSI. If you have an aftermarket pump, obviously, that does not apply and the manufacturer's recommendation for pressure should be used.
Pressure switches come in several ranges and there are some that you can individually adjust low and high pressure but they are expensive and not really necessary. the most common ones are something like 20/40, 30/50, 40/60. Typically, you can adjust the range somewhat but you can't vary the difference to any noticeable extent. IOW, you could adjust a 20/40 to say, 25/45 but you can't change the 20PSI range.
I suggest you check the tank to verify it will or will not hold pressure. If it does, and there is no change in operation, then I suggest you go to a 20/40 switch and after verifying the switch's/pump's cut-in pressure, set the accumulator to that pressure.