The deed is done!
I finally had the time to get down and dirty (literally!) and change the power supply. Well, the old one died on me and I had to replace it. Now the old 650W rests in peace while a brand new 850W does the job. And guess what, I was right. Replacing the power supply solved the “random” sleep problems of my server.
Finally I can watch TV, leave the office and do other stuff without having to fear that my server is in stand by.
And now to the next step. Get the OS moved to the new SSD. Not only a non-sleeping server, but also a fast one.
Although… my wife most likely to first clean the carpet…