snip

Very good suggestion. I had no idea about that relay and the possible
failure modes. I guess the best thing to do is replace the start cap and
that relay both and see if it cures the problem.

First choice would be that electrolytic start cap. Its the weakest link
in the system & its cheap to replace.

from a previous post: "The controller box is in the cellar, which
normally maintains around 66ºF, but on the very cold nights gets down to
60ºF. I'm wondering if the capacitors' electrical characteristics can
change that much because of temperature if the cap is marginal and on
the verge of failure."
Yes, it can change with temperature. If it changes from "just enough"
capacitance at 66° to "almost enough" capacitance at 60° the pump will
not start.

Still, at 20 years of age, even though it IS stainless steel, I wonder how
much life is left in the actual pump.

At 20 years replacing the start cap is a good idea even if it has not
failed (yet). The oil-filled run cap may last forever.
The start relay is fairly robust but might be going intermittent.
Replacing it now will not be a bad idea since it has had many starts
under its belt in those 20 years.

The same goes for the "reset overload protector". It seems to be
functioning correctly at the moment because it is tripping & you can
reset it - sometimes resulting in normal operation of pump. The typical
overload protector is a thermal device which "pops" when it gets hot
(due to overcurrent). Typical failure is tripping at a lower than rated
current due to aging & contact resistance. You may want to consider
replacing it even if it is not the root cause of your problem at the
moment (after you get the system back in operation) just to prevent
future problems.

How much life is left in the pump motor? It is hard to tell but each
time it trips the overload protector you are overloading the motor!
Typically the motor internal windings/connections get very hot quickly
if it does not start correctly. If given 5-10 minutes between each
attempted start they will have a chance to cool down due to thermal
conduction to the water. Repeated rapid attempts to start which result
in the overload tripping are bad. It the windings are already hot from a
previous overload & another attempt is started immediately they get even
hotter. Not good.

Enough for now... That missing ground does seems to need attention...
Please be careful

Kim