This applies to linear power supplies, particularly in antique (as in full of vacuum tubes) radio and TV sets. It is *not* advisable to operate anything with a switchmode power supply at reduced line voltage. The better designed ones will lock out and shut down, but some will crank up the duty cycle struggling to maintain output until something blows up.
In ~2 decades of working on this sort of stuff, I have *never* had a problem just powering something up. It is advisable to have a peek inside first to make sure it's not full of bugs, dead rodents, pee, water, metal filings, and other such things but assuming it passes the quick visual exam, plug it in, flip the switch, and be ready to kill the power if you see smoke or sparks. It's not a bad idea to replace caps that are known to fail, but unless they are obviously damaged, I normally check it out first, if it works perfectly, chances are the parts are fine.