I stopped doing PC repair in 2005 so I'm a little out of date (hence why I come here to learn a bit). First step was always identify the faulty peripheral or board (diagnosis by substitution), i.e. have power supplies, boards monitors and peripherals on the shelf. Same logic goes for any other system.
After that what the guy in the video says about asking questions saves a lot of time. Diagnosing totally 'blind' takes longer. Questions like when (thunder storm, hot weather), did they spill coffee on it, were they plugging/unplugging it, did it crack/flash/smoke/hiss, did their partner through it through the window just before it 'died', were they doing a firmware update that didn't finish, and in fact did it ever work or did they buy it as is from a boot sale?
Next is knowing (i guess from experience) what tends to go on certain things (common points of failure).
Power supplies are often fuses, bridge rectifiers, diodes, followed by voltage regulators and caps.
Motherboards used to have a lot of trouble with the caps next to the power connector (modern ones seem better), and really old ones (386/486) used to fail on cache ram, oscillators, keyboard serial decoders, and RTC modules.
Monitors (I can only talk about the old CRTs), were power switches about a third of the time, colour driver chip if it's lost 1 colour (usually LM1203N on the end of the tube), X2 capacitors near the flyback if it's got frame/field collapse, or the flyback itself if it's hissing. Also watch out for those fuses that look nothing like fuses (axial lead or small cylinders). You would need a high voltage prove and a high voltage discharge lead before tackling these things (26kv is fairly common at the anode).
Inkjet printers are usually power supply faults, laser printers usually just need cleaning or have failed optical sensors.
Modems were usually lightning strikes, and hard drives either head crashes or stalled motors (you can kick start them for data recovery by applying power then holding sideways and thumping it into the palm of your other hand). A stethoscope is handy for diagnosing HDDs as a rapid shortcut to the fault.
Most of the above requires little real understanding of electronics, from there you move onto more detailed diagnosis. I often found a logic probe as useful as a scope.
Perhaps the best use of time is to spend 5 minutes just looking at the board first. Probably half of the common faults show up visually. Of course if these boards haven't come straight from the consumer then the easy stuff has probably all been done, leaving you with the hard work.
Occasionally I'd have time for a 'proper' diagnosis, checking a board right through for the fault, but in IT retail the product is worth sod all and the customer only has £30 in his pocket anyway, so if I couldn't find the fault in 20 minutes I would swap the board. Hopefully you're not constrained by that sort of thing.