Once the tire's been inflated for a while it's stretched about as much as its going to. Mounting and unmounting doesn't stretch it.
Here's a rim cross section. The bead hooks are at the bottom. See how the 'floor' between the bead hooks is curved? If you squeeze the beads of the tire together they drop into the lower part of the curve. That makes the effective diameter of the rim smaller which gives you more slack for getting the bead over the edge of the rim.
When you're putting a tire on, put on one bead at a time. The first one will be easy. Then stuff the tube up under the tire. As you work the other bead on, stuff the tube up under the tire before you lever more of the bead on, so you don't pinch the tube. Once the tire gets tight I use the tire iron to stuff the tube up into the tire. Lever an inch or so of bead on, then do the next inch. When the tire gets tight go around the part that's already seated and squeeze the beads together into the center of the rim. You will get a little more slack that way.
You're trying to do it all at once, which is MUCH harder. And for gods sake stop using the screwdriver!
I never need more than one lever to mount a tire, and I have skinny climber arms. It's not about brute force, it's technique and paying attention.
When you are done, make sure that the tube is not caught under the bead- start at the valve and push the tire to one side so you can see if there is any tube sticking out. Then do the other side. Then move 4 inches and check the next section. It takes about 20 seconds if you're practiced and it will save you embarassment from a loud blowout- the bead will cut through the tube and the explosion will blow the tire off the rim.