They use ac voltage in which is different depending on the monitor and region and sometimes age (some 80's monitors stepped down to 100V from 120V via transformer.)
Most monitors require isolation.
Power supply doesn't power monitor. If you have 120V coming from the PS it is strictly pass-through. If monitor appears to have no power, check the AC voltage coming into the power connection. If you have 120 then start checking fuses.
Cap kits are easy but do take some time. Its just "busy work". I've done about 20 personally. They fix about 75% of monitor problems.
As I said, caps are the most common problem. Here is my troubleshooting step by step:
1-What is the monitor doing? Dead? Power but no pic? Distorted pic? Rolling pic? Vertical line only?
2-If no power, is there AC coming in? If so are the fuses ok? Check fuses with a meter not visually. If you're blowing fuses you have a short. It is probably a bad HOT, Flyback (check for cracks) or transistor though in rarer cases it could be a bad cap, diode or resistor.
3-power but no pic is there a signal coming in? Edge connector good? RGB sync cable plugged in correctly?
4-If distorted pic, rolling pic, vertical line and adjustment pots don't help: cap kit.
5-While doing a cap kit check for cold solder on on all header pins.
6-If after you cap the monitor you still have specific problems then break out the schematics and check the sections that would cause your specific problem. Example: If you have vertical collapse, check every semiconductor in the vertical output section. Start with transistors and diodes. If all test good start pulling resistors (not fun).
Some other notes. If after capping you can't sync still you may have bad pots. Failing flybacks can cause dark and out-of-focus pics (caps can also) KLOV forums are a great resource for troubleshooting individual monitors. Compared to pcb repair, monitor repair is easy and can save you tons of money. CRT monitors are getting scarce. Teaching yourself to fix monitors will extend the hobby for years. For example the Electrohome GO7 monitor is considered one of the best monitors still even though it is 30 years old!