I personally don't believe it's a sin to watch action movies.
For one, it's all stunts, pyrotechnics, stunt doubles, wire work, quibs (which are small controlled explosives for say flipping a car), etc. It's all completely fake and you know the main hero is going to come out of an otherwise impossible situation. It's silly, over the top, but in the end, it doesn't matter.
Also, think of some of the stories in the Bible. Some have completely brutal violence that would be rated 'R' if filmed exactly how the Bible portrays it, and no one under 18 could see it, yet we see little kids, like 8 yrs old, reading the Bible. How does that make sense?
If watching anything is a sin just because it has violence, or gore, or sex..then it would be (kind of) hypocritical to say reading the Bible is ok. I would also say that with Game of Thrones. I bet making an HBO-like production of The Bible would make it unsuitable for kids as well.