That is correct, when you wrap a stream in another stream, calling .Close() or .Dispose() on any of them will dispose the whole stream. In this case, wrapping the MemoryStream in a StreamWriter means that when the using statement completes the StreamWriter and MemoryStream are both disposed.
My guess is since you are returning a FileStreamResult the encapsulating File will close the stream for you after the stream is no longer used. In this case, you do not want to use the using statement and will want to leave the stream open when returning it.
UPDATE
Since a stream is forward access you’ll need to see the stream back to the beginning to allow the data to be read back out.
stream.Seek(0, SeekOrigin.Begin);