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...it doesn't mean they would suggest switching to your weak hand.
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I wouldn't suggest switching either.
But for someone brand new to shooting, I would think any professional trainer would at least explore the possibility of having a new shooter start out shooting using the dominant eye hand.
Switching is hard. When you're just starting out you don't have anything to unlearn so it's a completely different story.
The equilibrium issue is a good one. One of the best competitive rifle shooters of our time, David Tubb, has his rifles set up so that he can keep his head perfectly straight up while sighting his rifle. He eliminates even the small amount of head tilt that most people employ when shooting a rifle.
While it's more of an issue when precision is the goal, I would say that even in dynamic shooting one will find that performance is better if the head is in a reasonably natural position.
And in dynamic shooting, shooting with both eyes open is the goal. This is very difficult to do, perhaps impossible, when you are shooting cross-dominant.