That's a small example of what I mean, not the full totality of the threat space.
Building more readership makes them that much more difficult to cut funding to, in a variety of ways (more people to get outraged if the government tries, more evidence to show the government that they are worth the money, etc.). Clickbait headlines create more readership. We may prefer that to be false, but it's obviously not. From there the logic is simple, almost inexorable.
Being government funded does not mean you get to ignore funding issues, it just changes how you play the game. There is no source of unconditional funding in the world.
Also, one might argue, funding aside, that increasing readership (without sacrificing quality below some threshold) is the whole point of the enterprise. Click-bait headlines are annoying, but they aren't the actual body of the report themselves.
The BBC serves a charter which is managed by the BBC Trust. The Trust effectively interprets the charter. Their interpretation of the goals of the enterprise are (http://www.bbc.co.uk/bbctrust/):
"Make the most creative and distinctive output;
Innovate online to create a more personal BBC;
Serve all audiences; and
Improve value for money through a simpler, more efficient, and more open BBC. "
Increasing readership by reducing quality of headlines to match junk media clickbait doesn't appear to fit in there.
"The Public Purposes of the BBC are as follows—
(a) sustaining citizenship and civil society;
(b) promoting education and learning;
(c) stimulating creativity and cultural excellence;
(d) representing the UK, its nations, regions and communities;
(e) bringing the UK to the world and the world to the UK;
(f) in promoting its other purposes, helping to deliver to the public the benefit of emerging communications technologies and services and, in addition, taking a leading role in the switchover to digital television."
I'd argue that this sort of change goes against the stimulation of cultural excellence, the tail is wagging the dog.
It could reduce it, arguing there is a lot of waste at the bbc. As long as the public perceives the value in the bbc's offerings the odds of that happening are reduced.