There is no "seems" about it, the computer IS old. But the CPU is not fried from usage - if it was then it wouldn't even boot up. Hardware doesn't exactly slow down due to age. Software might, there was research done some years ago that showed that the older the OS installation is, the slower and more degraded it became to the point of "death" (AKA, needing to reinstall the OS).
Now new thermal paste isn't necessary so long as his computer isn't over heating. It might be, which can cause massive slow downs and may make the music stutter.
Also, that computer isn't even as old as you are. Not far off... but still.
And I wouldn't say its common sense. Perhaps he doesn't have the money to get a new PC? Maybe music was just now starting to do this. If it worked before, why bother getting something new? Just fix the old and keep on chugging.