I found that it is possible to clean Speakeasy's bilge pump--a Jabsco model that is still made with a different design. The pump and motor are separate. The pump can be taken apart where one finds various rubber, plastic, and metal parts and fasteners. On the bottom side of the pump is a rubber piece that separates the pump into two chambers similar to a reptile heart. On the top is the oscillating arm that moves the diaphragm up and down, which provides the pressure to move water. Also accessed from the top are two round metal and rubber pieces that move water in one chamber and out the other. These discs also do some straining of the bilge water.
I checked all of the pieces for tears, cracks, and wear. All of the pieces look surprisingly good. One can buy a new diaphragm or a more complete repair kit, but none of the parts appeared to need replacing. What I did find were three bits of plastic that had been sucked into the pump. Two of these bits were from tags on newly purchased clothing that had found itself in the bilge instead of in the trash. On a boat, cleanliness is safety.