You're not really "overcoming" hand dominance when you choose to shoot with your weaker hand but stronger eye. You're just training your weaker hand to do more things. Also, it's not that it's easy to train the weaker hand, it's just easier than trying to train your non-dom eye. the whole "just close the other eye" solution doesn't help if, like many people, your non-dom eye is also weaker and worse than your dom eye.
No, you are attempting to overcome hand dominance. Give a righty a wrong handed stick and see how it goes. Why is a rifle any different?
Have not found most of these comments or suggestions to be true across multiple people. As you mentioned, it would have a lot to do with the difference in eye strength, available coordination, etc.
For most encumbered with this issue, the easiest was to close the eye. It isn’t like the non-dominant eye doesn’t work. It certainly is the most practical, the least effort, and can still achieve results at the highest levels.
There is little value forcing someone to shoot wrong handed when a simple wink will solve the problem more times than it won’t.
Watched a fellow at the range attempting to do this with his kid as a direct result of forum “advice” Dad went out and purchased a left handed rifle for the kid, who was right handed left eye dominant. Kid would have been about 14. The whole show looked like a monkey humping a football. It was starting to get a bit heated. Asked a few questions. Dad had the same condition, but figured it was too late for him, and wanted the kid to do it as he had read about. Kid did everything right handed. Ask Dad if the kid could wink. Turns out he could! (Have never met anyone than can’t in under a minute). Kid shot better right than he did left, with the eye closed. Dad tried lefty as well and couldn’t get it to work.
This is one example of many. Multiple forums show the same results as above. They also show results where it worked or was “better”.
Again, if it works for you, great. That doesn’t mean it’s going to work for everyone else.
When the easiest thing to try is closing one eye, that should probably be the first step?
R.