Reland "[a11y-extensions] Refactor runWithLoadedTree to be async"

This is a reland of commit 83cfaf020896c2ed27cb11d36dbfb6e0cd94351a

There was one await missing, it has now been added.

Original change's description:
> [a11y-extensions] Refactor runWithLoadedTree to be async
>
> This change converts the test function runWithLoadedTree from a
> callback-based method to the ES6 async/await paradigm. This greatly
> improves the readability of tests, and the readability of
> runWithLoadedTree itself.
>
> The majority of this change is straightforward migration -- e.g.
> making the top-level function async, removing the callback braces, etc.
>
> This change is a pure refactor.
>
> AX-Relnotes: n/a.
> Bug: none
> Test: automated tests for chromevox, select to speak, switch access...
> Change-Id: If02ca57552f8cff9c1acad0e7d4d0587be08e2f1
> Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/3511256
> Commit-Queue: Anastasia Helfinstein <anastasi@google.com>
> Reviewed-by: Josiah Krutz <josiahk@google.com>
> Cr-Commit-Position: refs/heads/main@{#985573}

Bug: none
Change-Id: I630cb457b1f0ddc653535507185970cf956a96d0
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/3556126
Reviewed-by: Josiah Krutz <josiahk@google.com>
Commit-Queue: Anastasia Helfinstein <anastasi@google.com>
Cr-Commit-Position: refs/heads/main@{#986207}