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This is a re-land of ec6127ba.

The flakiness caused by the original CL doesn't happen after converting
WebTestHostMsg_WebTestRuntimeFlagsChanged.

Additionally, this CL removes unused WebTestMessageFilter classes.

Original change's description:
> Convert WebTestHostMsg_InitiateCaptureDump to Mojo
>
> This CL migrates the legacy InitiateCaptureDump IPC message
> to the new Mojo defined in WebTestClient interface.
>
> Bug: 1039247
> Change-Id: Idd38ecf67d10593f3bfc6f3c48d9832514cec595
> Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/2032634
> Commit-Queue: Gyuyoung Kim <gyuyoung@igalia.com>
> Reviewed-by: Dave Tapuska <dtapuska@chromium.org>
> Reviewed-by: Mike West <mkwst@chromium.org>
> Cr-Commit-Position: refs/heads/master@{#739222}

Bug: 1039247, 1049919
Change-Id: Ie58a28d6e6dddd26f59849da24579251ad7a970d
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/2043612
Commit-Queue: Gyuyoung Kim <gyuyoung@igalia.com>
Reviewed-by: Dave Tapuska <dtapuska@chromium.org>
Reviewed-by: Mike West <mkwst@chromium.org>
Cr-Commit-Position: refs/heads/master@{#747116}