Split SearchActivityUtils part 2: isolate Client logic (and tests).

There's no functional changes. The code is simply moved around.
Even static method continue to be static to simplify review.

Bug: 329663295
Change-Id: I325b686a9769d85c021794ecda6f0de19ee2633f
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5539511


Reviewed-by: Patrick Noland <pnoland@chromium.org>
Auto-Submit: Tomasz Wiszkowski <ender@google.com>
Commit-Queue: Tomasz Wiszkowski <ender@google.com>
Commit-Queue: Calder Kitagawa <ckitagawa@chromium.org>
Reviewed-by: Calder Kitagawa <ckitagawa@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1300863}