If everything is running fine, as you say, then a wipe and reinstall would seem to be overkill, when all that's wrong is… what? If the computer needed that icon there, then it would be unusable, don't you think? What if the poor 'puter, unable to eject itself, or to unbolt the drive, as instructed, merely did the next-best thing and 'copied' itself into the Trash?
Do not empty the Trash while you sort this out, but it seems this has not destroyed your ability to operate, so I think you'd be best to examine the situation further w/o taking drastic action.
For instance: I assume your Finder preference was set to display hard drives on the desktop (or you wouldn't have had anything to Trash). Is it doing so now, and if it is, what do you see when you double-click that icon? What isn't there, that you think should be there? If it isn't displayed, what happens when you toggle that Finder preference? What do you get when you do a Find on a filename like Library that the OS created? Or on your home folder? You can set Finder's new windows to open Computer, what do you see when you do?
If the computer needed that icon, it would have crashed, or not allowed you to do what you did, in fact displaying that icon would not be a user-defined preference (as it wasn't on OS9).
It may be that you have some work to do, but before doing the whole brain-transplant of a re-install, I think you should put a bit more time into diagnosis. The 'problem' may be smaller than you think.