[Cast Channel] Fix flags recording issue.

The Cast Authentication logic works by chaining several function calls
end to end.  We must copy the flags returned from a previous step to any
results from subsequent steps.

This also adds additional test coverage of the flags recording.

Bug: 1412904,b:239475260
Change-Id: Id4706924ff96cd53466a9786834e405458dd5de4
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/4348029


Commit-Queue: Mark Foltz <mfoltz@chromium.org>
Reviewed-by: Muyao Xu <muyaoxu@google.com>
Reviewed-by: Ryan Chung <ryanchung@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1119633}