One difference between the two systems that may or may not be relevant:
the one where this problem is occurring has a fingerprint reader, the
other one does not.
If I disable automatic login on the system having this problem and then
log in with my password, this problem does not occur. If on the other
hand with automatic login disabled I log in with my fingerprint, this
problem does occur.
So it appears that on the system where I'm not having this problem,
something in the login sequence is forcing GNOME to prompt for my
keyring password on login and that happens before gnome-remote-desktop
starts up, whereas on the system where I'm having this problem, that's
not happening so when gnome-remote-desktop starts up it can't get to the
password stored in the keyring to it creates a new, random password.
