Replacing all the parts just because "old" is not very economical, even if perhaps the safest thing to do. That being said time is also money so for anyone unwilling to invest all that effort into diagnosis and repair it might be better to just replace parts. Then again the problem with this approach is you learn nothing and if the actual problem has not been fixed you will blow up another set of parts, eventually, trying to get the PSU to run.
These are still pretty simple SMPS - there two sides, primary and secondary. Usually primary side should not die because of a short or overload on secondary side, and even some parts failure would not outright kill the PSU (for example in most cases you can't drive DC to the swiching transistors so it's not possible to turn them on permanently). But every time primary side dies it's a very good idea to check secondary as well.
Switching transistors can go open or short. Open is rare and more often then not is a result of a short that has vaporized parts of the smiconductor structure. So you need to look for what went open if the fuse didn't - the possible parts are the rectifier bridge (always check), the surge protection (some sort of NTC varistor usually) and possibly the coils on the noise filter. Sometimes the copper on the PCB can get vaporized but that is obvious to find and in general quite spectacular when it happens.
If there's 2 transistors usually both go, you can try and leave the other one if it tests good but frankly it might be a good idea to replace it, I would not trust it. With both transistors removed the entire input section, along with caps, can be easily tested - you just connect it to the mains via a 40-60W light bulb. The bulb should light up briefly as the caps charge and then go off. If not you have a short (could be the caps or the rectifier, or parts in the input filter). It's safe and perhaps even recommended to leave the caps reforming with the lightbulb for 30min, nothing should short or smoke, the circuit must be perfectly stable with no load. If the bulb is not on you can add miliamp meter in circuit, modern caps should have next too no leakage after reforming so the current should be below 1mA, if it's higer then either the filter or the electrolytics should probably be replaced because it'll only get worse.
NOTE: There are (or should be) resistors to discharge the primary side caps when there is no load, but that will take time. After you disconnect the PSU from mains do not touch anything on the PCB until you are sure the residual charge is low enough - test with a volt meter to be sure, the resistors could be open for example. Or missing because cheap PSU.
Once you are happy with the primary side, except the transistors, check the secondary side. Make sure all the replaced caps are soldered in correctly polarization wise. If in doubt locate the secondary GND (black wires) and go back from there. Check each rail for a short (note, some PSU can have below 100 ohm load on 5V or 12V lines, that is OK, a short is when you see below 10 ohms on a rail with nothing connected).
If everything checks out and you replace the switching transistors and the PSU dies again, then you can suspect the transformer or the PWM chip. And at this point you probably want another PSU since replacing the parts again is just a waste of money. Unless the PSU is somehow special (different voltages, or form factor) and you badly want it working again. BTW first PSU tests after the repair should also be done with the lightbulb (though 40W might be too small for an old PSU with poor efficiency) - it should prevent permanent damage. You try the PSU with no load and the bulb will be on but dim (or at least not full brightness). You can try up to 100W lightbulb and if that is not enough to get the PSU going by itself, you have more repairs to do.