Adequate programs log errors when things go wrong.
Solid programs also notify the admin when things go wrong,
and the programmer when insanity strikes.
No-one will actively monitor error logs just for the fun of it. Not for any length of time. So programs need to tell you when they put something in there.
And if the alert e-mails become a major hassle, the problem isn't the numerous e-mails, it's the broken program. Fix the problem (the broken program) to lose the hassle (the e-mail load).