Rules as written, throwing it would not accomplish anything. Despite being “delicate,” it turns out to not really be delicate at all from this perspective.
The reason why is Shalantha’s delicate disk is triggered when it takes 1 point of damage, and being thrown just doesn’t do that. Thrown weapons do not take damage for being thrown, or more broadly, weapons aren’t damaged by attacking with them. You have to deal damage to the disk to trigger it.
You could try, maybe, to toss the disk at a location and then use a second attack to hit the disk; its dimensions amount to probably a Tiny object (Player’s Handbook gives a book as an example of a Tiny-sized object), so its AC is just 5 (which is bizarre since the AC to hit a given 5-ft. square is also AC 5, and hitting a 1-ft. diameter disk should, ya know, be harder than that).
As for throwing the disk into the air and letting it fall, to take falling damage... who knows? That isn’t something that happens in the normal course of throwing things (and I don’t think anyone wants it to be, as it would be an enormous headache to play), so it would seem very strange to me for it to do so here just because you want it to. But, of course, that is inconsistent with how falling objects usually behave.
It does suggest that a flying creature could drop the disks as bombs, which would actually be a free action, which is quite something.
Ultimately, the disk is supremely problematic for all these reasons, and it’s probably best to just ignore it.