Fix: Restore download drag icon rendering

This is a follow-up CL to the previous CL [1], which moved the icon
painting logic back to `LabelButton`. The move was necessary as the
earlier fix CL [2] for this bug introduced visual regressions in other
`LabelButton`s, including some Ring 0 buttons.

However, bringing the logic back to `LabelButton` caused this previously
resolved bug to reappear — the icon stopped rendering when dragging an
item from the downloads bubble.

Fix: This CL disables icon layer painting logic specifically for the
drag contents button. This restores the expected drag icon behavior
without impacting other `LabelButton` instances.

[1] https://chromium-review.googlesource.com/c/chromium/src/+/6576165
[2] https://chromium-review.googlesource.com/c/chromium/src/+/6279834

Please find the attached screencast below after the fix:
https://drive.google.com/file/d/148X6lPYtpTQkhT-rtnmfR9wsYKbHKO4R/view?usp=sharing

Bug: 394380766, 414783688
Change-Id: I6584e1328072e7bb28bb47cfce76219f568d4ca4
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6580519
Reviewed-by: David Yeung <dayeung@chromium.org>
Commit-Queue: Vansh Parikh <vanshparikh@microsoft.com>
Reviewed-by: Ahmed Moussa <ahmedmoussa@google.com>
Cr-Commit-Position: refs/heads/main@{#1464503}