Even as a kid decades ago, I taught myself how to add and subtract using what today people would call the common core way.
It made no sense to me, when given the question, what is 999 + 2001 to start the process in the one's column, carrying one, etc. I started with the most significant digit and worked left to right. It scales beautifully, and it gave me a sense of the size of the numbers. If I made a mistake, I'd be off by one or two. I'd love to see the example cited in your post.
There are bad math teachers. Perhaps your third grader has one. Perhaps the teacher doesn't have a good grasp on what they're teaching. Perhaps they're like you and don't want to change.
It's possible that kids learn differently and some love math while others don't. Some kids probably grasp the common core method slowly but would have excelled at the old way. Teach them the old way.
And have your kids take "high stakes testing." Only parents think of these tests as high stakes. The kids' lives don't change based on any single test. Even the SAT can be retaken. As for me, I had my kids take the SAT when they were in middle school. They did ok. But they did a lot better the next time, and the time after that.