A pop through the speakers is a good sign. I would, first, remove the chassis and look over all connections. Look for burnt resistors, bulging capacitors, broken wires, wire out of a socket, etc. If all looks good, I would remove all the tubes (make sure you know where they go) and spray all tube sockets and the backs of any switches, especially the power switch as they are known to go bad, with TV tuner cleaner or electronic switch cleaner. Turn or move all the switches and controls, put the tubes in the sockets, then remove them again and spray everything again and move the controls again. What you need to spray are the contacts of the switches and controls. Most pots (like a volume control) have an open area where the wires come out, that's where you spray. Then reassemble it and see what it does, if anything. Make sure to check the schematic or tube chart to be sure the tubes are in the correct sockets. Remember, someone was probably in there before you and could have mixed them up.
The rectifier issue is a common problem, so are bad capacitors, especially the filter capacitors for the power supply, but the set would usually hum, even with bad caps. I wouldn't rule out the tubes, but they are tougher than most people usually give them credit for. Make sure all of them are lit. In my experience with German sets, they have a lot more issues with resistors and wire breaks than American sets. Also changing the rectifier, talked about previously, is a great idea as they are not reliable once they age past a certain point. Pretty easy repair too. German sets are also not the easiest things to work on, BUT when you get them to work they sound better than anything in their vintage and they look beautiful.