Hopefully this page will help anyone else who wants to get their clock running again.
Once I knew the trick to fix the clock, it was actually a relatively trivial matter. Essentially, the potting compound (glue?) that Sony used to hold down the quartz oscillator used for time keeping purposes had become somewhat conductive due to age or heat or something. This then prevented the time keeping circuit from working.
To fix the problem, one simply has to remove as much of the old glue as possible, clean the board up, and at least in my case, re-flow the solder for the quartz oscillator, and you're done!
Honestly, the hardest part is getting the screen re-connected to reassemble the clock.
The fix for this was originally described by 'gunter.fritz' on the Sony UK site here:
https://community.sony.co.uk/t5/hi-fi-systems/dream-machine-icf-cl75ip-cant-change-date/td-p/652659
Check out my step-by-step instructions on how to fix this yourself.
I have only owned my ICF-CL75iP for a few weeks. Everything is working so far (fingers crossed), except the battery indicator (battery with a slash through it) is on. I put a brand new CR2032 battery in and followed the instructions to make the indicator go away (p. 29) but no change. Any suggestions? Thanks!