Alright, I've got the machine here now and have tried it out. I started with the original power supply and the condition is the same:
As discussed in PM, we decided to make a fresh power supply in any case. The C64 needs 5 volts of DC and 9 volts AC, which is a peculiar combination and you sure can't find a single power supply these days which provides both. You can find separate supplies of course.
The 9 volt line needs to provide over 1 amp of current, and I couldn't easily find a supply like that (only some with 0.3 amps or so). However, 12 volt linear DC power supplies are easy to find.
The way these work is there is a transformer to drop down the mains voltage and a
rectifier and
smoothing capacitor to make the output direct current. It just so happens that the AC output of the transformer is 9 volts, which is exactly what we need. So I simply took out the rectifier diodes and capacitor. The output voltage turned out to be a little bit higher than that of the original PSU, but this should not be a problem.
(The C64 motherboard in fact rectifies the AC to 12V DC - just like the PSU circuit did - it would be possible to bypass this and just feed in 12V DC, with some caveats, but this is a topic for another thread.)
The 5V power supply is a switch type device which means it doesn't need a bulky transformer and can be quite small, sort of like most mobile phone chargers are these days.
I combined both power supplies with a suitable Y-cable arrangement and a fresh 7 pin DIN plug, and it works well. Less elegant than a single brick but hey, at least it's brand new.
Will start replacing the old RAM with some sockets and new memory chips next.