This sounds incredibly dubious. As an iOS dev myself, there is no way for me to continually record a customers screen, UIKit wouldn't allow an open ended session like this nor would it 'capture' without notification.
What's more likely happening is that the app records it's inputs and somewhere (call centre perhaps) is capable of simulating this playback. If you're using Apples native keyboard, they cannot see your key presses when the keyboard is visible as this would be a protected and completely different process than the app recording inputs.
I'm not surprised that this news article has no video evidence of this alleged screen capture happening because it's simply not possible with the current 3rd party development toolset.
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If only you lived closer to me, I'd love to demonstrate showing you how much more secure browsing on your iOS device really is; you'd be blown away. iOS has a bounty program though where Apple pay fat stacks of cash to those that properly disclose and document vulnerabilities. Mac OS on the other hand? No such thing. One might ask themselves why this is and perhaps also why Apple has Mac computers right in its iOS sights. As for Windows well, they're better than they used to be lets go that far. My advice, stick with iOS. It has the biggest amount of cash behind it and is arguably the best OS on the planet.
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I'm fairly certain Apple are doing what you expect them to do. Remember, every process that is built by a 3rd party for iOS must conform to the constraints of the 3rd party developer framework. We simply cannot write code outside of that. If the API does not support 'continuous screen recording without customers explicit consent' it just cannot happen and I assure you (until I see evidence which non of us have yet) it cannot happen unless some exploit has been found which in itself is worth a lot more money from Apple Security than it is from a news agency for the story.
