In other countries, people are used to receiving text messages after each of their credit card transactions. Given that this helps alert people to possible fraud, I'm surprised credit card companies haven't pushed for this to be an opt-out system, especially since most already allow you to setup your own alerts; it's just not the default.
The main downside is that this potentially results in an onerous number of text messages, that the credit card company would have to pay for (likely at a fixed rate), which would hurt their margin on transactions. But, I have a hard time believing that this would be cost-inefficient for credit card companies who'd likely need to investigate fewer chargebacks as a result. It's easier to stop suspicious activity right when it starts instead of trying to piece together when a card was compromised.