Thanks for explaining that to me. YOur description of how her attention wanders constantly was clarifying. It does seem to back up the notion that she does not see you as a VIP. She may be one of those horses that has a very short attention span, and this requires you to constantly keep pulling it back from her. AND, to improve on that, you may have to work with her is such a way that she takes a good, long second look at her owner and sees someone who is worth looking at. That would mean maybe working with her in round pen or on a long lunge and makeing her move (someone said lots of changes of directions) and making her LOOK at you . You can't blame her for having a short attention span. But you need to be at least part of what she is looking at. Then when her attention wanders, you interrupt it (not punish!) so that it becomes "noisy and uncomfortable" when she is looking elsewhere, and quiet and peaceful when she looks at you.
I agree that riding in the side of the road now sounds risky, but you sound like a tough cookie to me. ME, I am probably old and overly cautious in comparison.
One thing, please try not to get into the mind set of thinking like your horse is behaving this way just to be "bad" or to get your goat. Once you set up an adversarial way of thinking toward your horse, you will have many negative feelings that will only interfere with your ability to make yourself a GOOD place to be and all those other things not worth the bother.