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Calling JS functions by name from C++ is an obsolete pattern. Use
FireWebUIListener/addWebUIListener which is the suggested approach.

Also extracting all chrome.send() calls to a local BrowserProxy class
which makes the code more readable and is consistent with the
BrowserProxy pattern used in other WebUIs.

Bug: None
Change-Id: I911b114b6687e7b0108296a38636811bc066dd12
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/2450722
Commit-Queue: dpapad <dpapad@chromium.org>
Reviewed-by: Dominic Mazzoni <dmazzoni@chromium.org>
Cr-Commit-Position: refs/heads/master@{#814634}