Have you calculated the additional fuel burn on an annual basis of the unneeded additional weight?
The piece of paper weighs the same and burns the same amount of fuel whether it’s in the glove box or in his wallet in the drivers seat.
Yes, but I propose carrying the whole 8.5x11 sheet and @kay8907 had proposed carrying just the ~2x3" card.
If we take those dimensions as accurate, I’m carrying 93.5 square inches of paper versus 6.
I have.
But then I opened the window while driving and stuck my tongue out like a dog for 12 seconds, raised the window, and had pissed away a year’s savings with the smaller lighter paper. 
Do what I do-fold the 8.5” x11” into thirds as if to mail it, as you described. Then tear off the top third along the crease and put that in your glove box-not worth futzing with scissors to cut it down to exactly 2” x 3”. Actually, in my state, there is a back and front of a 2” x3” card printed on that 8.5 x 11 sheet so to get both required sides you have 2 x3 twice with the inch strip between them so 14 square inches at a minimum vs 93.5” with @LSLGuy’s method. My compromise of fold, crease and tear off top third of 93.5 gets it down to 31.2 square inches, barely more than the 14 square “ with all the futzing. A non-futzing fuel wasting I can live with.
I print it on onionskin paper to minimize weight.
And just how many sheets do you wind up sacrificing to printer jams? 
Because I was taught you never leave your registration in the car, & for the same reasons, you shouldn’t leave your insurance card there either. If it’s freshly stolen & they get pulled over & reach into the glove box & have all of the paperwork the cop is either gonna write them a ticket or give them a warning & off they go w/o another thought by the cop; just a routine traffic stop, soon to be forgotten. However, if they don’t have any paperwork & don’t even know the name of the owner, which is printed on the insurance card, along with the owner’s address most cops will do a look a bit closer at that situation. At the very least they may remember who was driving it when it does pop up on the stolen list.
That being said, I kept the registration in my wallet (until the state stopped sending them out; now you’re supposed to print it yourself) & the insurance in the car mainly because those cards didn’t fit in the wallet, even folder in ½ they were wider than a standard credit card/license slot
As for me, I want a paper insurance card rather than on my phone. I'm not handing my unlocked phone over to a stranger who might then take it away out of my sight.
My registration is a full 8.5x11, I’m not keeping that in my wallet, especially now that I carry a card-only wallet. On top of that, my wife and kids also drive the car.
As to the cards, Auto-Owners always sent two cards for each vehicle (I usually have three). One card for each vehicle’s glove box and one each for my wallet. Mercury insurance did the same, but with the OP’s plain paper style. This go round I changed to Progressive. They do it differently. Each vehicle is listed on each of the four plain paper cards, so I end up with one for each vehicle and one only for the wallet.