My question: I know painting a metal cabinet is a no-no.
Say what? Maybe you know something I don't, but a good paint job can cover a multitude of sins. The operative concept here is "good paint job". Where most people go off the skids is with a lack of proper surface prep, the patience to primer & level the base coat before applying the color top coat, and a poor choice of paint gloss & color. I like shiny new things as well as the next person, but on an old radio a low-gloss matte finish plays much better. If the first thing you notice about a restored radio is the paint job there is an issue with the paint job. You want good surface prep, uniform color, and nothing that distracts from the more attractive features of the radio. Think autumnal umbers and warm greys, avoid hammertone green whenever possible.
If your arm isn't sore after prepping a cabinet you aren't finished prepping the cabinet.
But what about wood? Is the value/worth of the radio lessened by a re-finish job?
This requires a judgment call... Antique purists avoid the words "wear & tear" and came up with "patina" as a polite code phrase for "wear & tear". Sometimes it's much easier to spin chipped & scratched veneer as a sign of character in a 70 year old patina than it is to refinish the wood in a way that says "nice" without saying "over restored".
Some of the old cabinets had beautiful wood grains and colors that are so dulled and oxidized it takes a skilled eye to appreciate them, which is exactly why it's a good idea to develop your wood restoration skills. If the casual observer takes note of how nice the wood looks that means it was properly restored. You want depth & clarity in the wood grain and color so avoid anything that plugs up the surface texture. Polyurethane and Shellac are marvelous for table tops and floors, but on a radio cabinet I much prefer Minwax stains & oils. You can build depth with each coat, it won't chip like Tung Oil will, and you'll find a variety of colors at your local handy guy store. On some cabinets Minwax Natural (clear) is more than adequate to refresh the wood, at other times a light Walnut can add needed color to faded wood. The slight yellow of an Oak stain can warm up the veneer in a way that says 'old' without saying 'fake'. Much of this is a judgment call that comes with experience, careful consideration of the condition of the radio, and the Zen of the radio itself... Some consoles bordered on the gaudy with dark stain accents while other areas featured cross-grain veneer work that deserved a clear / light tint oil to make them more obvious.
If you're sensitive to the style of the radio itself that can tell you what to do, and sometimes it pays to go slow and mull it over at length.
...how would you fill in a couple of minor chips? Would you use a wood filler?
Filler putty is quick and easy but has no grain and will never blend as well as a good veneer patch. Working with veneer is both a skill and an art that requires practice. Anyone with a block of wood and some 220 grit sandpaper can level out filler, but what makes a patch blend is matching the grain and cutting the patch to go with the flow of the piece. Avoid square patches, go with oblong shapes instead. If it's a small ding a veneer patch will likely be much larger and that might be a consideration, if the radio has a multitude of dings it might be worth stripping off the old veneer entirely and starting fresh. I have an old Philco with badly faded & cracked veneer on a sound cabinet that's a good candidate for a veneer transplant. I should also mention that only a bit of glue is need to hold a veneer patch as even a light surface treatment like a Minwax oil will penetrate the patch and bond the pieces together.
Always remember that you can't permanently screw up a metal cabinet because no matter how bad a paint job might be, a week of drying followed by an hour of sanding can bring it back for a second chance. With a wooden cabinet you'll typically block sand it well enough to remove surface oxidation and level the texture, but not much more as removing too much wood creates problems for the next restoration........