The camera already has a protective glass cover over the lcd that can be easily replaced. A protector to protect the protector?
Easily? Seriously? I think you're exaggerating that a bit. I replaced the *plastic* screen on the 5D classic and it was anything but easy. First you have to buy the part directly from Canon support, not the most intuitive thing to do. (Be sure to order the adhesive as well, since they don't include it with the screen!) Then you have to heat up and pry off the old screen without scratching the rest of your body or worse, the LCD itself. It might technically be "easy" to do, but it's nerve-wracking, having your unprotected LCD screen exposed to the elements. The risk of damaging the LCD is not worth it to me.
And I think you're wrong, but I'm having trouble proving it... The LCD screen does not appear to be actual glass. It feels like the same soft plastic that's on every other Canon body I've owned. They scratch incredibly easy. Why Canon refuses to put something like Gorilla Glass on their pro DSLR's is beyond me.
Save yourself the trouble. Use screen protection films. They really do work, and XO skins are as smooth as glass instead of "orange peel" textured, so the reflections will not look weird.