We played around with old gear like this back in the late 70's. Vintage speakers connected to modern stereo, modern speakers connected to vintage amps, that kind of thing. I had built a Dynaco Stereo 400 but didn't really know anything about how to modify circuits or even swap out parts like caps to improve the sound.
Based on what I know now here's what I would do if I had a vintage console: as little as possible. Most of these things are pretty simple. Great big solid wood box with simple circuits with point to point soldered wiring in an open sheet metal chassis is pretty common. The speaker(s) are always paper cone stamped frame cheap looking things which nevertheless if you hook them up to something modern you will be surprised how good they sound. I would take them out but unless damaged would put them right back in.
With everything out I would carefully refinish the cabinet. Use Fine Woodworking or other trusted source for what to use and how to do that. Huge amount of the charm is in the appearance. When in doubt just clean it up, give it an oil rub, call it good. But there are woodworking techniques for restoring or recreating original finish. This is at least as important as anything you do electrically.
Most of these things have the sound coming from a solid wood opening with fancy patterning with fabric with more fancy patterning behind it. Cleaning that fabric or finding authentic replacement could be one of the trickier parts of the whole thing.
Then simply go through the electrics and clean, test or simply replace tubes. Might redo solder but I wouldn't re-wire, at least not unless you can source original looking wire. Half the cool factor is the old school look!
Then I would put it all back together with one modification, either new quick-release fasteners or and open back. You want to be able to see and show off all your great work.