Migrate HeadlessDevToolsClientImpl to base::Value::Dict

This CL reduces the use of legacy `base::Value` APIs in the
implementation of `HeadlessDevToolsClientImpl` in favor of the
`base::Value::Dict`. As a consequence, some of the methods could be
simplified as we no longer need to pass the owning message around,
and we should also be able to use const-references until a copy is
actually needed.

Bug: 1187038
Change-Id: I9da6f3823cf53067b72b8f99d06370f9866e04b5
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/4102843
Commit-Queue: Piotr Bialecki <bialpio@chromium.org>
Reviewed-by: Andrey Kosyakov <caseq@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1083352}