Ok, replaced the cap with a molded Siemens film cap 200 nF / 350 V that measured 205 nF salvaged from an old fluorescent light bulb. Turns on right away, maybe a tad brighter than before. High-pitched noise seems a lot better. Cannot hear it from driver's position but then it is raining all day. Noise is still there when I bring my ear close to the radio. It seems a little more broadband now.
Measuring with a phone app and the phone held right into the CD slot, the main components of 7.9 kHz and 15.8 kHz are still there (about 40 and 30 dB above noise level). I remember they were much stronger and could be measured from 10 - 20 cm away. Gone is the sporadic peak at 9.9 kHz, but instead there is a whole carpet of changing peaks between the fundamental and the 2nd harmonic.
Now, unless I accidentally chose exactly the same capacity as before, why didn't the inverter frequency change? I cannot find any IC that injects a frequency near this power supply.