The "official" troubleshooting methods work reliably because they make few assumptions, and so you should follow them in the general case. However, if you're looking for shortcuts that work more often than they have a right to, here you go: The most common failure in modern supplies (so-called "switching" supplies) is of one or more electrolytic capacitors. Quite frequently, a simple visual inspection suffices to detect failure -- there will be noticeable bulging on top (and/or perhaps on the bottom).
Also check for a blown fuse and perhaps a cooked resistor (again, you can detect these failures with ease); replace if necessary. Replace the defective capacitor(s), and you'll often be up and running.
Troubleshooting other cases involves more work and more sophisticated test gear, but the "method" outlined above will frequently suffice.