[CodeHealth] Cleanup remaining ListValue usage in site_settings UT

Removes deprecated ListValue usage in the site settings handler and
helper UTs. This also changes some of the product method signatures,
but data is exposed via callbacks which can still wrap the new types
in a base::Value before sending them onwards.

Migrating off of ListValue allows moving off of the v1 version of the
Append APIs and on to the v2 versions of those APIs.

Bug: 1303949
Change-Id: I27e58550ef085326e7cb355866b15c3df1ba37ec
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/3756912
Auto-Submit: Alexander Cooper <alcooper@chromium.org>
Reviewed-by: Theodore Olsauskas-Warren <sauski@google.com>
Commit-Queue: Theodore Olsauskas-Warren <sauski@google.com>
Cr-Commit-Position: refs/heads/main@{#1023615}