I took a look at the reproduction project to see what you were trying to accomplish here...
To my understanding:
XCTAssertThrowsError are assertions that takes in a block that can throw. They just happen to assert that an error is thrown in a synchronous block when it's done running.
XCTestExpectation are classes that keep track of whether or not requested conditions are met. They are for keeping track of asynchronous code behavior objects/references need to be kept and checked later.
What you seem to be trying to do is make something like XCTestExpectation work the same way XCTAssertThrowsError does, as in make an synchronous assertion that an asynchronous block will throw. It won't work quite that way because of how the code runs and returns.
The asynchronous code you refer to does not throw (timer initializer). As far as I know, there aren't any asynchronous blocks that can throw. Perhaps the question you should be asking is how can we make a synchronous operation choose to run synchronously sometimes, but also asynchronously when it feels like...
Alternatively for some additional complexity in every class you would like to test I've made a solution with what is almost bare minimum to make this easily testable and portable...
https://github.com/Jasperav/ThrowingAsyncError/pull/1/files
May I ask why you would ever want to do something like this?