Calling async method from sync method is ASP .NET web api - issue

I have a ASP .NET web api 2 application, and I'm trying to call asnyc method from sync method and meanwhile I encountered some issues. If I just call the method as a regular method the body after delay doesn't get executed, if I call it with Task.Run() it gets executed

public void CallingMethod(MethodExecutionArgs args)
{
  //do something with the args
            System.Diagnostics.Debug.WriteLine("BEFORE ");

            WriteFileToDiskAsync(args); // If I run only this, I never see "WriteFile() - AFTER delay" in the output
            Task.Run(async () => await WriteFileToDiskAsync(args)); // this executes the entire async method
            System.Diagnostics.Debug.WriteLine($"Finally written from sync method");

}

private async Task<bool> WriteFileToDiskAsync(dynamic file)
{
            System.Diagnostics.Debug.WriteLine("Before delay inside async");
            await Task.Delay(3000);
            System.Diagnostics.Debug.WriteLine("WriteFile() - AFTER delay");
}

Result: I always see the lines written from CallingMethod, but when I call the async method like a regular method, I only see the "Before delay inside async", If I call it with Task.Run() it see both lines from the async method.

Question1: Can someone explain this behavior, why doesn't the async method execute fully? Does it have something to do with what Stephen Cleary says: If you lose your AppDomain for any reason, that in-progress work is lost.

I read these articles but can't figure out why is this happening:

• How to call asynchronous method from synchronous method in C#?
• Fire and forget async method in asp.net mvc
• C# async/await with/without awaiting (fire and forget)
• https://blog.stephencleary.com/2012/12/returning-early-from-aspnet-requests.html According to most articles what I'm trying to do is not recommended (among other things the exceptions from the async method would get swallowed) but in my case I already have the information that I need to generate the response and what is done in the Async method doesn't concern the client at all...

Context: What I'm trying to achieve is, on a existing API route that creates a X Resource (saves it in a database), after the X resource is created, I want to asynchronously call a method that will Create a json file with some information from that X Resource and save it in a filesystem. But even if the writing of the file fails, I want to return a successful response to the client (because his request was to actually save the X Resource - which succeeded) he doesn't care about Creating external file in XYZ filesystem.

Question2: What would you say is the best practice to achieve what I described above, considering all the existing methods chained in the CreateResource route are sync?