Use a separate channel to report the final error of the copying
goroutines, receiving a value only when all of the copying goroutines
have completed. In a followup change (CL 401835), that will allow
waiters to select on goroutine completion alongside other events (such
as Context cancellation).
Also mildly refactor the watchCtx helper method so that its structure
better matches what will be needed to implement WaitDelay.