Now exactly the same problem has returned! By visiting Yamada Radio Clinic's archive and reading his article ALC Problem with TS-850 => DC-DC mod I learned that the transistors in the oscillator can degrade over time due to overload. This is probably what happened to my old TS-850 transceiver, which was produced circa 1990.
The new DC-DC board for Kenwood TS-850.
I decided to replace the old oscillator board with a new board using modern components. I followed the idea from Yamada Radio Clinic and made a replacement PCB containing the voltage converter ICL7660 and three more components. The result was satisfying and my Kenwood TS-850 now runs "just like new"!
If you want all details, you can visit my homepage where I've described the task and added more photos and a circuit diagram.