Being part of the right-handed race I can't speak from first hand experience...
However, from trg the lefties drills when forced to use a non-left handed weapon are slightly more complicated. In order to keep their master-hand on the weapon they have to rotate the weapon to an unnatural degree to reach the cocking handle, and then they reach over the weapon and action. Sounds easy, but as soon as you add in all sorts of maliciously located objects with sharp edges such as an ACOG, night aiming device, torch, spare rails etc they regularly risk any potential future as a hand-model with the increased chances of cutting, gouging or otherwise disfiguring their fingers, the poor blighters!
Training everyone to fire right-handed may be a good thing - I have no qualification to comment, and have heard arguments go each way. I like the idea of simplicity and the get-on-with-it approach your describe - after all it is muscle memory, and a well enforced drill that is 80% efficient will be infinitely better than 3 variations of the same drill to achieve the elusive 100%.
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