Mercury 1998 8HP 2-stroke, primer diaphragm problem - ?

It's a manual start, long shaft. Well used. We've had a few things go wrong
over the years....but, in general it has been an incredibly reliable engine.

After a couple of strokes on the primer knob, it has always started on the first pull.
Now, it doesn't. The linkages do seem to be depressing the diaphragm button.
I'm guessing the diaphragm itself is torn. The question is: am I going to have
to take the carburetor out of the engine to remove the diaphragm cover and replace
the diaphragm?? If so, I may wait until the boat comes out of the water at the
end of the season....

It's easier if you do.

Under the cover there is the little (plunger) sticking up that's integral to the bladder. It's what gets depressed by the primer lever (finger) when you pull the choke knob. There is also a gasket and a spring and a little ball bearing/little spring in there (plus a couple of other bits that slip my mind at the moment).

The bearing is vital to the proper operation of the carb in general and it can be a booger to get seated properly. (if it pops out of where it's supposed to be you will flood the engine when you try and run it).

So while you could probably remove enough of the primer assembly (the mechanical parts) to get at the pieces that are on the carb, it's probably not something you want to work on in "close quarters" like that.

On a total carb rebuild on these models if anything is going to have to be "looked at again" and fixed due to something getting out of whack on reassembly it's going to be the primer.

Personally, I would wait until you can pull the carb and do get the primer repair kit (about 5 or 6 pcs including the bladder).