Why do people talk about being overburdened by politics, religious controversies, etc.?

It's one thing to be worn out because you've done a lot of work. It's another thing to be worn out because you've done a small amount of work while being out of shape.

When it comes to matters like religion and politics, modern Americans (and whoever else) are clearly in the latter category, not the former. For a large amount of documentation to that effect, see here.

Since Americans have false priorities, they're more concerned about politics than religion. I'll start with politics. One of the most common complaints you hear in political contexts is that people are tired of all of the political ads they see on television during a campaign season. Of course, to see a lot of television ads, you have to be watching a lot of television. And what people are watching is typically highly secular and/or trivial (and often sinful). The reason why political campaigns run so many television ads and do other such things is that they know people spend so much time watching certain television shows, listening to certain radio programs, and so on. And the ads are shallow because the audience is shallow. If you're hundreds of pounds overweight, you might complain that leaning over to pick up something you knocked on the floor is an excessive burden. But whose fault is it that you're so overweight and have been for so long? The fact that leaning over and picking something up wears you out doesn't prove that it's asking too much to expect you to do it. The same people who are irritated by having their television shows interrupted by political ads are also highly ignorant of their nation's history, their nation's political system, political issues, etc., and they're culpable for that ignorance.

The situation is even worse with religion, which is more important than politics and is even more neglected. Pastors, apologists, and others who are interacting with people on religious issues need to study the culture they're interacting with (like with the resources on the page I linked earlier) and make adjustments accordingly. When you hear about how people supposedly are worn out in religious contexts, are tired of all of the arguments about theology, are overburdened by apologetics, and so forth, you need to keep in mind that it's more often than not a matter of people being worn out by doing a small amount of work while they're out of shape. They need to get in better shape and start doing a lot more work, not less of it. Not only does far more work need to be done, but the neglect on the part of the large majority of people is placing too much of a burden on the small minority who are doing too much of the work.