I got a new diaphragm, but looking at the old diaphragm, it looks to
be intact and in good shape. The material still seems pliable, I held
it up to a light and can't find any breaks in it, including where the
choke plate rod is attached.

The longer of the two pickup tubes had a fair amount of crud on the
screen, which I've cleaned off. Gonna put it back together with the
old diaphragm and see if it starts any easier with that pickup cleaned
off. If it does, I'll store the new diaphragm in the refrigerator for
future use.

And why are there two pickup tubes and why are they different lengths?