Prevent ThreadContext header leak when sending response backport(#68649)

[pgomulka]

We need to stash thread context in DefaultRestChannel before we call
channel.sendResponse because when calling this method, our execution
might be delayed and the thread be reused for another task - like
sending another response.
And it would see thread context from the initial “delayed” work.

This commit also expands the assertions on the empty thread context
to make sure it does not contains response headers.

closes #68278

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[elasticmachine]

Pinging @elastic/es-core-infra (Team:Core/Infra)