Fix Issues with CS Handling in ILM Async Steps (#68361)

* No need to spin on all CS changes when waiting for a snapshot to go away,
just use an appropriate predicate
* Stop waiting for a new state if we are not master any longer since we can't
complete the action anyway at this point (worse yet it will potentially run concurrently
on local and master if it uses the `client` to execute its requests)
* Properly fail the CS listener if the node is shutting down to get easier to debug
logging
* Stop throwing exceptions in `clusterStateProcessed` in the error step mover, this is forbidden