nless the Ring of Amara works off the demon possessing the corpse, in which case it would either do nothing or destroy him.
If you find it, you can have a magic user study it. Maybe it's worth using, maybe it's not. But at least you checked it off the list. Also why the fuck would it do anything if it worked on a demon, it didn't just randomly kill people that put it on... so why would it kill the Paul like zombie? Run some tests, have Giles call his aunts and take a look see... the other side of the coin is, maybe it works and you have a second line of defense. The character wouldn't know which side the ring would fall on until he gets it.
(Also time loops like what happened in the Buffy show are a thing... so you can with the help of a friendly magic user... TOTALLY TEST IT! Without risking your life...)
Hour Man's steroids and Thinker's cap which needed to be plugged directly into his brain and couldn't be safely removed.
There were other heroes/villains and Alan in this is already partially something else... so more the rest of the lanterns. But okay... let's go with the Thinker's cap. Ring scan the tech, see if you can get something that doesn't need to be plugged in. It might be worth handing the design off to a friend or it might not be..
And you're assuming that he could find a ring based on the description 'a ring somewhere in Sunnydale'.
No, I'm not assuming he can find a ring based off it's a ring... It's in a vault, underground with a lot of gold... so yes, I think a fucking Power Ring can scan for that... (considering your were using a Power Ring to mine asteroids which would require scanning at 100's or thousands of miles, through all sorts of crap...) if you find another vault with gold... you loot it and keep looking. So yeah, I think you can find it. It's also in the main town area based on the collapsed sections of road. I mean, boosting recall shouldn't be out of the question for Paul, thus... hey, it's underground in a vault with gold... something that would be useful for him anyways...
So... yeah, I don't really understand a reason not to look considering you'd just have the ring make a map and ask yourself, is this large stash of gold in the bank... no, okay, let's check it out. It's an action that also gets you gold and potentially other things, it's not a oh woe is me, I spent months or years looking for this and it didn't pay off perfectly...
This is a hey, I spent a few hours, maybe a few days and I got a magical item that can be studied and possibly replicated for use by Angel if you can't use it yourself. Or maybe you can figure out how to create rings that stop sunburn... that's still worth a couple of hours or days when it also comes with vault filled with gold.