Use ClipboardData and WriteClipboardData instead of ClipboardWriter

This matches clipboard history's mechanism to copy things to clipboard,
and allows us to share the same `ClipboardData` when inserting media via
pasting (see https://crrev.com/c/5736869).

Bug: b:351050139, b:337064111
Change-Id: I6a6a210004586ae43df462c624620c5e75590807
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5737069
Commit-Queue: Michael Cui <mlcui@google.com>
Reviewed-by: Darren Shen <shend@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1332225}