Fix "Re-starting finished log context" spam when creating events (#14947)

`run_in_background` calls re-use the current logging context. When they
are not awaited, they can complete after the current logging context has
been marked as finished, which leads to log spam. Use
`run_as_background_process` instead.

Fixes one of the instances of #13090.

Signed-off-by: Sean Quah <seanq@matrix.org>