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Currently, the callback is passed to SourceStreamToDataPipe in the
constructor.  This means that it has to be owned externally while the
callback is outstanding.  By moving the callback to Start, the
SourceStreamToDataPipe can be owned by the callback itself.

https://chromium-review.googlesource.com/c/chromium/src/+/1830360 uses
this logic.  Breaking this small patch out from there.

Change-Id: I93ad30f243e29f61e903dbd6b4ddfa37652e05c0
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/1841184
Auto-Submit: enne <enne@chromium.org>
Commit-Queue: Kinuko Yasuda <kinuko@chromium.org>
Reviewed-by: Kunihiko Sakamoto <ksakamoto@chromium.org>
Reviewed-by: Kinuko Yasuda <kinuko@chromium.org>
Cr-Commit-Position: refs/heads/master@{#703582}