Can a fuel pump get weaker and slowly fail? I had always heard either they work or they don't. And no, there is no gas leaking or weeping from it.

Yes it can. Mechanical fuel pumps have two valves inside, one to prevent it from sending fuel back to the tank while pumping to the carburetor, and another to prevent it from sucking fuel from the carb when getting full from fuel from the tank, basically. If one of the two starts to leak, the pump's action will suffer until it finally gives up on you. That is known as an internal leak.
When gasoline is leaking from the little hole on the pump, that's because the diaphragm is gone, producing a external leak. To the engine, this last one is worse, because the gas will contaminate the oil.

On the old pumps, the ones you could disassemble, you could buy a rebuild kit, that came with new valves and diaphragm. Some even came with a new arm and spring.

Túlio Lazzaroni "FORD", Florianópolis SC Brasil.

'74 Ford Galaxie 500 292 V8

'82 Ford Galaxie Landau 302 V8

'98 Chevrolet S10 4.3 V6

'01 Ford Focus 1.8 Zetec

I had a valve come out of the housing, so that under normal driving it would run OK but if I put my foot down it would starve for fuel. What made it more puzzling is that it did not do it all the time as the valve would drop back into its seat and start working OK.

Don't forget things that could simulate a slowly failing fuel pump - such as restriction on the suction side like a kinked suction line, a restrictive suction filter, restriction at the suction line inlet inside the tank, contaminates inside the suction line ; or restriction on the discharge side such as kinked discharge line, restrictive add on discharge filter, restrictive ceramic or screen filter inside the carburetor.