Destruction of the The Ring...

[Amarie of the Vanyar]

Going back to the original question of why the destroying of the Ring causes the end of Sauron, the tale The Faithful Stone in the Unfinished tales, explains a similar case.
In it, Aghan, a Drúedain, passes some of his power to a watch-stone. The basis of the stone is burned by fire, and as a consequence the legs of Aghan blister. After thinking about it, Aghan says:

If some power passes from you to a thing that you have made, then you must take a share in its hurts

The Faithful Stone; Unfinished Tales