[Read Aloud] Move Uma time recordings to use chrome.metricsPrivate

This reduces the number of things we're forwarding to the controller and
can be done easily with chrome.metricsPrivate. This uses the new Logger
class and MetricsBrowserProxy.

Bug: 346612365
Change-Id: I15f935985b5d53af58b61b3f23bcfd6c695c08f7
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5631491
Reviewed-by: Abigail Klein <abigailbklein@google.com>
Commit-Queue: Kristi Saney <kristislee@google.com>
Reviewed-by: Demetrios Papadopoulos <dpapad@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1317584}