Yes, he should call you. He should charge you enough money that it's worth his time to do that.
(I had a hosting service that my credit card started bouncing on delete some data a few years ago. I was furious. I'd paid them a bunch of money over the years, and they knew my phone number: if they'd called me after I didn't respond to their emails I'd still be their customer, and I'd still have that data).
For a backup service, I'd be perfectly happy for it to hold on to my encrypted data for a year after I stop paying, then charge me a hefty "recovery" fee (at least how much I should have paid for the time that I wasn't, and quite happily more).
The comparison is flawed. Would be better to ask what if you leave your dog for 6 months in a dog-hotel and the 7th month you don't show up?
I'm sure your dog will end up dead eventually or nowhere to be found.
That said I get your point. But rolling your own notification solution shouldn't be that much of a problem for technical people or people with deep pockets. Just hire a programmer to write an application that reads picodollars and if the predefined threshold is passed the program makes a phonecall/sends 15 emails/call the local authorities?!.
