I've heard this idea before but I think it's primarily hindsight. At the time The X Files was on, there were even more popular shows about strong institutions, like Law & Order or The West Wing. So if popular TV was influential, why didn't society evolve toward those portrayals? I think we can look back today and say "that looked like this," but it doesn't mean that caused this.

It took something far more powerful than a TV show that was mildly popular for a few years to create what we live today. It took the great flatness of the Internet. As Terry Pratchett predicted to Bill Gates back then.