Can I just clarify what you do and see? You see a hit come up in the search box, click on download and watch it download to a completion? But when you look in Library, it is not there? Further, knowing the name of the artists or songs, Finder cannot locate any MP3 matching what was downloaded? All that would be a strange scenario.

The launch button only works if you have an internal MP3 player and that player is set as the default format for downloads from the internet. As a matter of interest, what internal MP3 player does your OS9 something machine have in its bundle or have you subsequently downloaded? I know that, for example, the excellent iTunes did not become available until early 2001. If you acquired your machine before then, it might not have the best MP3 player aboard. I can give you the URL for a download if you can identify which OS you have.

When you give us more information, we can better advise.