Internal: b/144488201

To reproduce:

1. Run webview_flutter/example.
2. Tap into a text field. Observe that the keyboard is shown as expected.
3. Tap anywhere else on the screen.

Expected: Keyboard minimizes.
Actual: Keyboard persists.

Haven't tested this on iOS, but this reproduces on all versions of Android. This means that it should be unrelated to any of the ThreadedInputProxyAdapterView logic and may be a deeper problem with the platform views. The keyboard also appears to switch from the text field's InputConnection. If there's any text typed in the field the suggestion bar will lose it, and it will switch from title to lower case.

See the reproduction also in #36250.

(/cc @amirh)