I'm usually starting with a visual inspection inside the device, looking for swollen caps, leaking batteries, discolored/burnt PCB laminate, physical damage, foreign objects, etc, and general cleaning. If there are fans, I'm making sure they rotate freely.
Then it's time for the capacitor reforming mentioned above, focusing on large caps exposed to abuse in the power delivery chain, like smoothing caps in the PSU, voltage regulators, etc. Failing signal conditioning caps seldom or never lead to catastrophic failure. It's best done by desoldering the caps and reforming them individually out of circuit, but guess that would be quite impractical when dealing with a large collection. It might be more feasible to simplify it by doing it with a voltage adjustable AC supply instead, that easily could be designed and built around a triac, if there's nothing available commercially.
If possible, I disconnect and start the power supply unloaded, checking its output quality, verifying voltage(s), looking for excessive ripple etc.
Once that's done, I reconnect the PSU and power up normally, but keep an eye on temperatures etc for a while, making sure there's no runaway anywhere. A thermal imaging camera is immensely helpful. Nowadays there are Android phones with that function built in if you're on a budget. I'm also verifying the PSU's output quality loaded.
I also like the idea of using an additional smaller fuse earlier in the thread. I'd get an assortment of B and Z curve circuit breakers in small sizes, say 1, 2, 4, 6 and 10A and choose among them according to the device's rated current. It might limit the damage done if there's a problem.
Also, keep in mind that there are probably as many methods of doing this as there are electronics geeks, not trying to suggest that my method is perfect in any way 😀
Oh, and be ready for cracks, pops, smoke, arcing, and even fire. Always think it through and prepare with an action plan before you power anything up. You might also want to use some PPE, I've for instance had one of those through-hole mounted fuses from Littelfuse literally explode in my face. Good thing I had glasses on. The bang gave me ringing in the ears for hours afterwards.