Today, scheduleTask to does not execute the scheduled task to completion, when the task is a future. If we are providing a Future<int> Function() to scheduleTask, the current implementation of scheduleTask returns a future when this task starts, but not when it completes. To actually run the task to completion, the callsite needs to look something like the following, which is kind of awkward:

void foo() async {
  Future<int> Function() task = () async => 1;

  // Awkward?
  await await binding.scheduleTask(task, ...);
}

I stumbled across this when using the API above but with a single await, and I was quite confused why the unawaited_futures lint was triggering which led me to looking deeper at the issue.

It would be desirable for task to be scheduled to completion, even when task is a function that returns a future for a simpler API. We can already opt-into this behavior "for free" when the future is handed to completer. The signature of Completer.complete is:

void complete(
  [FutureOr<T>? value]
)

From the docs (emphasis mine):

[case 1] Completes future with the supplied values.
...
[case 2] If the value is itself a future, the completer will wait for that future to complete, and complete with the same result, whether it is a success or an error.

Today, we're hitting the first case, but with the correct type annotations in this PR, we could get the second.

I also tested this change in an internal presubmit (tap/476354531), and it passes tests so it doesn't look like a breaking change.

Pre-launch Checklist

• I read the Contributor Guide and followed the process outlined there for submitting PRs.
• I read the Tree Hygiene wiki page, which explains my responsibilities.
• I read and followed the Flutter Style Guide, including Features we expect every widget to implement.
• I signed the CLA.
• I listed at least one issue that this PR fixes in the description above.
• I updated/added relevant documentation (doc comments with ///).
• I added new tests to check the change I am making, or this PR is test-exempt.
• All existing and new tests are passing.

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