[Code Health] Migrate e/c/manifest.cc to base::Value::Dict interface

There are still some leftovers related to the external callers. They
will be fixed in follow-ups

Bug: 1366865, 1187061
Change-Id: I76e4bf9d88a2a4de4bb26cedc7d1786fce08f4de
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/4025927


Reviewed-by: Reilly Grant <reillyg@chromium.org>
Commit-Queue: Roman Sorokin <rsorokin@google.com>
Commit-Queue: Jeroen Dhollander <jeroendh@google.com>
Reviewed-by: Jeroen Dhollander <jeroendh@google.com>
Auto-Submit: Roman Sorokin <rsorokin@google.com>
Cr-Commit-Position: refs/heads/main@{#1072146}