Please check this thread: https://discussions.unity.com/t/async-and-uncaught-exceptions/824272/28 for more info, in particular:
- issue only happens when running async method from coroutine `yield return MyMethodAsync();`
- GetAwaiter().GetResult() trick can suffer thread pool starvation and deadlock (freezing Unity)

I don't know the exact issue described in this ticket, but void async, as well as non awaited Awaitable are not logging Exception, and this is the behaviour to be expected, as the Exception is trapped by the Awaitable, and thus never to be logged by anyone.

One correct way to get exceptions logged is to use async event methods ( async Start, async Awake, etc. ) like so:

async Awaitable TestAsync ()
{
throw new Exception ( "Test of exception" );
await Awaitable.NextFrameAsync ();
}

protected async void Start ()
{
await TestAsync (); // exception is caught & logged by Unity code that started async Start
}

Another way, if called in a non async method, is to do :

protected void Start ()
{
TestAsync ().GetAwaiter ().GetResult (); // GetResult executes PropagateExceptionAndRelease which raise the exception trapped in the Awaitable
}

Finally, you could, but I don't recommend, use an extension method like:

public static class AwaitableExtensions
{
public static Exception GetException ( this Awaitable awaitable )
{
var edi = awaitable.GetFieldValue<ExceptionDispatchInfo> ( "_exceptionToRethrow" );
return edi.SourceException;
}
}

...
var asyncOpe = TestAsync (); // exception is kept in the Awaitable, and won't be logged
...
if ( asyncOpen.IsCompleted ) // exception also "complete" the Awaitable
{
var e = asyncOpe.GetException();
if ( e != null ) throw e; // propagate it yourself
}
...

TaskScheduler.UnobservedTaskException solution is not working with Awaitable.

In case anybody needs a temporary solution until unity fixes it.
The following code should should catch these exceptions.

System.Threading.Tasks.TaskScheduler.UnobservedTaskException += (_, e) => Debug.LogException(e.Exception);