[Code health] Avoid Value::GetAsDictionary (in /chrome/browser/net)

This method is deprecated, so port away from it, along with other nearby deprecated base::Value APIs that can be modernized at least somewhat locally.

This CL was uploaded by git cl split.

R=wub@chromium.org

Bug: 1187011
Change-Id: Iafc798d913f8e4eb631f9f19226289f77519f470
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/3826745
Commit-Queue: Bin Wu <wub@chromium.org>
Auto-Submit: Maks Orlovich <morlovich@chromium.org>
Reviewed-by: Bin Wu <wub@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1033993}