I have read the complete original Gizmodo article and that is not the implication that is being made.
Implications != reality. When an article is full of weasel words like "potentially", watch out! It's exactly the same kind of clickbait that we see about Android "viruses".
First off, you left out the most important sentence in the article:
"
Alternatively, it’s possible that Apple sandboxed the entitlement to prevent it from accessing data outside Uber’s app."
If Apple did that... and it does seems likely that an iOS API would indeed be limited only to its own space, as Apple does that a lot... then all the other conjecture about looking at other apps is totally bogus and it's a non-story.
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As for the idea that someone could break into Uber's network and somehow get access to everyone's iPhone, well think about that.
What they're implying is that someone could hack Uber's source code and put in some code in their app so it would record and send screens to a third party server... without anyone noticing the code had changed.
Trouble is, companies like that use source control. So there'd be a record of a change, and the change would likely require a developer's password to commit.
Secondly, if we're going to bring up scenarios like that, a similar kind of hack would apply to ANY iPhone app. Gosh, somebody could break into Wells Fargo's developer network and modify the banking app to send them our bank passwords. Ditto for keyboard apps.