When you say it never goes to the desktop screen, does it actually do anything else unusual ? Furthermore, does it do anything differently with the hard disk drive disconnected (don't do this if there is more than one disk drive).
(obviously this would not be my advice if I could access the bios myself.)
Broken hard drive is more likely than "broken" OS, all hard drives break. If the hard drive is worn out, they're only £30 to replace.
The OS being unlicensed is unlikely to have anything to do with anything.
That you have a 2nd computer can make this all the more easier to fix, as you can read about the motherboard while at the computer. When you say you tried installing a new OS, how far did you get? Could the advice in the post above help?
As for "not" doomed, you're going to toss a computer because one of the cheapest parts appears to not work?