[CodeHealth] base::Value modernisation for FeedbackPrivateDelegate

This CL modernises all uses of base::Value around
FeedbackPrivateDelegate, removing all deprecated methods in use.

Bug: 1187001
Change-Id: I43b0e50a0cb055fe80ca913ad0de59a0c26701e6
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/3878582


Reviewed-by: Istiaque Ahmed <lazyboy@chromium.org>
Reviewed-by: Giovanni Ortuno Urquidi <ortuno@chromium.org>
Reviewed-by: Colin Blundell <blundell@chromium.org>
Commit-Queue: Claudio DeSouza <cdesouza@igalia.com>
Cr-Commit-Position: refs/heads/main@{#1045023}