Needless verbosity is the problem here, IMO. I’d expect to receive output from the installer only when the download or install fails for some reason, not when it succeeds.

(Also, to me the culprit here seems to be flashplugin-installer and not update-notifier, as plenty of other packages also report their installation progress, but only flashplugin’s seems to get forwarded to root’s mail even when nothing abnormal occurs.)