Is that it's only function?
Yes. It's major function was to alert all concerned of any critical situation on a particular version of the network, same way people are sent warning triggers when a natural disasters is expected to hit.
Would catastrophe ensue if an evil person got ahold of the bitcoin alert key or a copy of it?
That was possible. It was also becoming difficult to manage due to emergence of different types of wallets.
I still don't quite grasp what it is, however.
Like, is it a thing? A button? A script? Part of the software itself?
Does it have a private key? Can someone hold it in their hand or does it only exist in a state of information?
It's a private key which was given to Bitcoin core developers at the time -
https://en.bitcoin.it/wiki/Alert_systemThis is another reason it was stopped, cause they could give it, but not take it away. Past developers who were no longer actively working on the network still had access to it and could send a alert message at anytime without anyone knowing who exactly sent it.