Make some InterceptedRequest methods private.

There were a couple of methods that were public when they could have
been private, also rearranged them in the file so that they're roughly
in the order they're called. Finally removed GetCookieStringOnUI method
that didn't seem to be used or even defined.

Bug: 447580987
Change-Id: I5b03368162f6a016dc37ecb500866181e3479912
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6994309
Reviewed-by: Nate Fischer <ntfschr@chromium.org>
Commit-Queue: Peter Conn <peconn@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1522699}