I typically go in the garage, open the door with either
the button or the remote, pull the car out of the garage,
then attempt to close the door with the remote.

The door closes OK about 50% of the time. When it doesn't
close properly, I can work the remote any number of times
having no effect. But if I wait 2 min. until the light goes
out, actuation with the remote works fine. A new battery in
the remote has no effect on this behavior. Positioning of
the antenna wire also has no effect.

So I guess this is a "weird wiring" question. Does anyone
have any idea why actuating the door with the remote would
be dependent on the light-timer circuit?

Take out the bulb... Does that make a difference? Could be current leakage
from a failing component. Might even just be RF noise drowning out the
remote from a poor contact on the bulb someplace.

Does the hardwired button work when the remote doesn't?

It might not even be related to the timer... The trigger circuit might just
happen to require about the same two minutes to recover from whateve causes
the failure.