Yes, but I believe in a society where almost all of the major news outlets are owned by supporters of the current government, the politicians also have a strong ability to control what their voters are demanding.
But for certain issues they line up behind the government. The Snowden files story was initially fully covered only by the Guardian -- who have since had a change of Editor after their offices were raided by the security services and equipment smashed. There has been (until now) almost no coverage that the government wants to pass the snoopers charter soon. There is no coverage that the 'emergency' DRIP bill from last year is due for renewal before Christmas. Due to the way that DRIP was rammed through Parliament last year in defiance of normal procedure my MP promised me a robust debate before the renewal. It looks like it will go through without so much as whisper.
EDITED as the telegraph currently has a front page article on the snoopers charter on its website. I hope that there is time for people to lobby their MPs (which is what killed it off last time).
As for Snowden, even on this website where people actually understand the potential of surveillance techniques it's the stories about his personal situation and options, as widely run in all UK newspapers from all angles, that seem to have captured the imagination more than the details of what was revealed. And I'm not sure he and his allies, who sought to carefully control how PRISM was reported, even offered the files to other UK newspapers...