...to my own question is, "yes."
Here's what I found out:
After doing the Special Secret Nut Calesthentics and ripping the poor toilet out by the roots, I started looking around inside. What I noticed was that several of the hoses were very, very old and nasty looking. Plus, they looked like they had stretched on the ends and were loose. I replaced all the hoses with new vinyl and double-clamped the pressure connections at the valve and the other end of that hose.
So far (knock on wood) no leaks!