[OmniboxEverywhere] Fix threading issues in the screenshot capture flow

Fixes several bugs and in the desktop screenshot capture flow:

a. Removes the IO thread task runner: CaptureScreenshot already forwards
the work to a worker thread.

b. Fixes a potential race condition: we were setting `is_capturing_ =
false` too early in OnScreenshotCaptured(), before the "processed
phase".

c. Fixes a bug in CaptureWorker::StopOnBackgroundThread:
DesktopCapturer::Start(nullptr) triggers a DCHECK due to the nullptr;
this wasn't caught because the MacOS DesktopCapturer handles nullptr but
the webrtc one used on Windows doesn't. Simply resetting `capturer_` is
sufficient.

d. Wraps the response callback using
mojo::WrapCallbackWithDefaultInvokeIfNotRun to eliminate use-after-move
and ensure Promise resolution if capture fails synchronously.

Bug: b:532198850
Change-Id: I08c3f12bb06f393a619b6fc2ccb9153b8d1846ee
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/8253215
Reviewed-by: Tove Petersson <tovep@chromium.org>
Reviewed-by: Hamzah Behery <behery@chromium.org>
Reviewed-by: Juliet Lévesque Knighton <julietlevesque@google.com>
Commit-Queue: Kaan Alsan <alsan@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1683893}