Apply cluster states in system context

[DaveCTurner]

Today cluster states are sometimes (rarely) applied in the default context
rather than system context, which means that any appliers which capture their
contexts cannot do things like remote transport actions when security is
enabled.

There are at least two ways that we end up applying the cluster state in the
default context:

1. locally applying a cluster state that indicates that the master has failed
2. the elected master times out while waiting for a response from another node

This commit ensures that cluster states are always applied in the system
context.

Mitigates #53751
Rework of #53785

[elasticmachine]

Pinging @elastic/es-distributed (:Distributed/Cluster Coordination)

[DaveCTurner]

This reworks #53785 for the 7.x branch since the backport of the original PR highlighted a problem with deprecation warning headers.

[DaveCTurner]

Closing this as it's not a good long-term solution and the preferred short-term solution is #53838.