I appreciate the concern, but my problem is more on my ability to concentrate on what's in front of me. I'll be walking around or climbing up stairs while my brain is in la la land thinking about something else. That's normally how I end up clutzing out.
The logic in your post is sound though, I never did think about my core.
But I do work out and I do core workouts alongside arm day and leg day. If anything my core is the thing I'm strongest in.
That makes sense. Our subconscious is better at picking up the small details than our conscious mind, because it takes more effort for us to think about it. That's why they say "go with your gut" because our brains calculate things subconsciously. It's why we can do things without thinking about it.