It already happens here. Transactions are done through USSD, rather than apps (though there are apps for them). The one big downside is that you can't send or receive payments anonymously. Your full names as they appear on your ID comes up with every transaction you make. Though, it is hard to track/keep data other than number+name.
But I think it's because of the regulations here, and perhaps in the EU they might make apps and regulations that make it possible with privacy.