If you have ever taken one of these pumps apart you will see that the diaphragm is a very thin piece of material subject to age & dry rot. Kits are available for some pumps. Might as well get a new pump. Your time is worth more than messing with the thing past making sure there are no air leaks in connections on the low pressure side of the pump.
Just went down the failed pump road. Leaking gasket on the pump. I carry a new spare so the change out only took a few minutes. Since then I have hopefully repaired the old pump by replacing the gasket & it now sits ready to go if needed.