I just think people's perception in how easily their opinion should change public policy is wrong. And I'm glad it is. They usually don't know all the nuances behind a given policy.

I see people both irritated with the lack of change in American government while also not politically active (Facebook doesn't count).

Creating policy change is challenging in a government defined by federalism and representative democracy like the United States. I agree letting someone know is likely not enough to make a big change (and it shouldn't be). But if enough people think one way and let the right people know, then they can influence change. People need to organize. A large group that agrees on a topic has influence. Elected officials will listen to them because they can move a large number of votes. The group needs a way to communicate their desires and power (ie. numbers) to elected officials. This is usually accomplished via interest groups which can lobby for their interest. I know lobbying is a dirty word, but it exists on one end of the spectrum. The spectrum exists from attending a local town hall and voicing your solo opinion (which I recommend doing!) to smaller organizations on local issues all the way to the major interest groups (Sierra Club, NAACP, NRA, AARP, etc) voicing millions of opinions.

There is a correlation, but not on the magnitude per person, work required, or time frame that most people hope for. You likely won't make a meaningful change with a single letter (I say likely because you can find counterexamples).