gh-114914: Avoid keeping dead StreamWriter alive (#115661)

gh-114914: Avoid keeping dead StreamWriter alive (#115661)

In some cases we might cause a StreamWriter to stay alive even when the
application has dropped all references to it. This prevents us from
doing automatical cleanup, and complaining that the StreamWriter wasn't
properly closed.

Fortunately, the extra reference was never actually used for anything so
we can just drop it.