Improve use of thread context propagation in CCR

[DaveCTurner]

Today CCR sends its cross-cluster requests using a client which switches thread context to one with the right authorisation headers just before sending the request to the remote cluster (see e.g. CcrLicenseChecker#wrapClient). Instead we should run all the relevant code in the right thread context to begin with, and then just switch to a regular client. This would prevent surprises such as #61308 and #84409 and would also let us propagate other headers across clusters (e.g. for tracing).

[elasticsearchmachine]

Pinging @elastic/es-distributed (Team:Distributed)