Just the opposite occurs. The less people pay attention, the more government intrudes into their lives.
I'm not really convinced of that. As recently as the 1970s, when there were only 3 TV networks, no internet or cable or home-video to distract people, and everyone got their news from Walter Cronkite, and arguably the goings-on in politics dominated peoples' thoughts to a far greater degree than they do now, government was quite involved in peoples' lives in ways that are somewhat unthinkable today.
Wage and price controls for example, implemented by Republican President Nixon.