While neither my eyes nor my brain are as sharp as they once were, at least they still match my right-hand shooting form. So I've never actually had off-eye-dominant issues, but I've experimented w/ the concept.
Even if we (correctly) don't focus on the bead during a swing/shot, the bead is definitely involved in a gun mount. When learning/practicing a mount, we judge it by where the bead floats in relation to the rib or receiver. (Also, we relate it to gun fit. How many of us pick a pretty, new gun up off the rack at the store, mount it, & then close our left eye to see how it "fits"?) Ideally, we practice enough that a good mount becomes second nature. But still, while focusing on a bird during a shot, the brain knows the bead is there & uses it, either to establish lead or to judge the quality of my mount. Occasionally, I'll be able to correct a mount prior to pulling the trigger (almost subconsciously), not because I'm checking the bead, but because my brain still sees it & determines that my mount's not perfect. Sometimes I'll miss a shot & immediately know it's because my mount wasn't right.
I'm pretty sure the concept between the Eyemaster, the tape on the glasses, etc. is the same. It's not so much about checking the bead (which takes focus off the bird & can lead to aiming). But by either moving the bead, or by taking the dominant eye out of the equation, your brain is tricked into believing your dominant eye (if you have one) is looking straight down the barrel, rather than across the barrel.
For over-thinking nerds like me, this is fun stuff to think about.