I can give you a definitive answer to the question in title. However, you have an unusual situation, and it's only possible to speculate as to what else it could be.
Could your non-pure sine wave UPS be responsible for the damage?
The waveform from the UPS cannot cause damage. If your computer has an active PFC power supply, a cheap non-sine wave UPS can lead to the equipment shutting down at the time of the switch-over, but not damage. If the UPS worked fine in keeping things powered up during outages, you don't have to worry about the waveform issue or need an expensive pure sine wave UPS.
Also consider that some of what failed was never plugged into the UPS.
Could the problem be due to power surges?
Cheap surge protectors that use MOVs are single-use protection; the MOVs sacrifice themselves and tend to burn out in order to protect your equipment. After they have done that once, the protection is usually lost.
That said, power surges short of a lightning strike tend to be self-limiting. For example, if the computer's power supply is not able to handle the surge, it is likely to fail before the components it powers are damaged. If you were to sustain surge damage beyond the power supply, it is likely that most components would fail simultaneously. It doesn't sound like that has been your experience.
If your utility power is horrendous, you might benefit from the type of UPS that is always active. Rather than switching over to battery when the power goes out, you always run from the battery and the UPS keeps it charged. That isolates your equipment from the utility power.
However, if the utility power isn't adequately protected from lightning, even that might not be enough to protect against a serious lightning strike. In fact, there isn't much that will, short of unplugging things. But again, the failure pattern you describe doesn't sound like lightning or surges.
The multiple failures aren't from the UPS, but they sure are unusual. My first reaction might be that the failures were misdiagnosed and some of the early replacements might not have been necessary. If the problem cleared up after replacement and some time later there was a new problem, that would indicate that this isn't the case. Although..., the process of servicing the computer might have included blowing it out or other actions that temporarily solved a different problem, like heat.
If the computer failures are all caused by a single underlying problem, there are only a few common denominators. One is the internal power supply, another is heat.
The only way I could envision the power supply damaging many different components would be if the voltages it is putting out are too high. I have never personally seen a power supply do that, but it's something you could check with a voltmeter.
If the inside of the computer is getting too hot, or heat isn't being dissipated you could get failures, although some components tend to shut down before they are destroyed. Signs to look for would be air vents blocked with dust, or exhaust fans caked with dust. You could also get excessive heat if the computer is in a confined space.
Multiple component failures might be due to bad computer build quality. Your computer could have been made on a Friday out of all marginal parts. Except you mention two AC transformers, a modem, and its power pack all failing, also. Unless you buy only cheap stuff from disreputable merchants, it doesn't seem likely that everything would fail, and do so in a short time frame.
Coming full circle
Which points back to a power problem, perhaps successive power surges on your utility line. Each surge might weaken various items; not enough to cause immediate failure but failure after varying numbers of similar hits. You could get a condition like this if you are next to a commercial or industrial facility with heavy equipment cycling on and off. If your only protection is old surge protectors with fried MOVs, you might not have any protection. Of course, replacing those with other MOV-based surge protectors puts you back in the same situation after the next surge. The computer component failures don't sound like a typical surge problem, as I described earlier, but your whole situation is unusual, so it's hard to rule anything out.
It might be worth talking to your power company. They may be able to put monitoring equipment on the line temporarily to see if there is a problem.
"Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth." Arthur Conan Doyle
Of course, just because something is improbable, it doesn't mean it can't happen. Somewhere in the world, somebody will be the one person who experiences every component failing in quick succession. You might be that person. Or at least there might be more than one underlying problem.