As the power section is now working, I re-checked all the wires I had to re-solder during the physical cleaning and decided to go for an "all-in" approach. That is in this case "all tubes in". Since the FM input path goes right through the AM path, I figured: if that works, I can pretty much go on. But I never figured that it actually might work...
I put back all the tubes and since two of them are incredibly similar to each other, I had to resort to some high level game of "spot the difference" to find out which tube goes into which socket. Oh, did I forget to mention that the print on the tubes wiped off togehter with all the dust?
Here we go: tubes - check. connections - check. Makeshift antenna - check.
I honestly didn't expect anything particular when turning it on, I just wanted to make sure there was no "magic smoke" and all the fillaments light up. Imagine my surprise when I heard static on the speaker... so I decided to manually push the FM tuning rod. MUSIC! Sweet, sweet music!
So as of now, I can pretty much declare that restoration a success. Needs quite some tuning still, but hey... I was listening to the news just now!
Right now I am not so sure if the AM part works on its own, since I don't have any antenna that would work, but hey, I am optimistic!