Comment by procaryote

You hash the servers because then adding or removing a server doesn't directly affect other servers position on the ring; adding a server just takes some load from som servers.

This is useful because you want stickiness, so requests for the same key mostly go to the same server.

Sorting servers by weight means that removing or adding a server will shift a lot of traffic from the servers it used to go to. A flapping server early in the list will break stickiness for the whole set of servers.

The simplicity of stable hashing means you don't have to think about new sets, old sets, table rebuilds, synchronisation schemes etc, and that's useful because every such extra step adds bugs and corner cases