How does anyone go about "rationally"
analyzing such problems?

See the cracked solder spots.

But that's part of the issue. A close visual inspection can be useful, but
it consumes a lot of time and is not "rational" -- that is, it doesn't
connect symptoms with possible causes. It's more an act of desperation.
(That is not an implied criticism. I'm a great believer in the "try
anything" approach to servicing, and often have to fall back on it myself.)