Is the issue really that "hide IME" is only called in specific
circumstances (e.g. focus change), and all the other IME-hiding behaviours
are implemented by the Content embedder based on other signals?

Yes, there is no explicit signal from contents to the embedder to "hide IME". The blimp client explicitly hides the IME right after sending the text to the content layer. But for the new UI specs, we would show a progress spinner and dismiss it only when the client gets a HIDE_IME signal back from engine. (Spinner CL in progress : https://codereview.chromium.org/2393443004/)
In this CL, we send a fake signal to the client with some delay. This can be improved by listening to a more appropriate event from contents.

Adding tracking blocker, moving out milestone based on priorities.

Obsolete, WontFix.