"How much time did you save by not typing my, and how much time has been spent finding problems caused by a typo in a variable name?"

^ That.

I have typed out use strict; at least several hundred thousand times over my 20 years writing Perl, and if it saved me even just 10 minutes troubleshooting why a variable is being clobbered, or why a variable isn't updated as expected, I feel my efforts were worth it. Also, by forcing you to declare all variables, it makes it far easier at a glance to see where variables are designed to be scoped. Without declaration, it's hard to tell whether a variable is global or not without having to scroll through code. That's even more time saved.