That's a nice idea too, but in reality it creates trouble (I tried it once). For example, my bank's fraud department often likes calling me several times a day asking me to "verify" transactions. I told them several times to NEVER call me, and that I will monitor my transactions via the online interface and call in the event of suspected fraud. However, they continued calling me, often at the most inconvenient times. Like when I have a bag slung across my shoulder, a grocery bag in one hand, a hot drink in the other, and standing on crowded public transit, they expect me to answer my phone, spill my drink, fall over on a moving train, and yell my social security number in front of everyone on the train? To top that, don't they realize that I need to login to my AWS account and check my instance usage before I can "verify" any transactions? Sorry bank, just DO NOT CALL. Ever. Period. E-mail me. This is the 21st century.
So back when I had one of those error message tones, they once decided that my contact information was "out of date" and just disabled my credit card while I was eating lunch, leaving me stranded at a restaurant like a fool, ATM and credit cards all disabled, without any way to pay. (Like seriously, are you trying to stop fraud, or stop your own customers?)