If my bank sent me a text message or an email every time I paid for something, then I would know immediately if there were fraudulent transactions on my account.
Downside is that this is a lot of text messages / emails, but not really. Not for me. I probably pay for stuff once every couple of days on average (Maybe I'm more miserly than most people). Could cut down on traffic with some rules e.g. transactions >£10 or only payments at a new merchant.
The real reason they don't do this, is because banks profit from people being very "fluid" with their money. They don't want to do anything which pours cold water on that.