You stripped the screw head because you pushed beyond the capacity of the screw. In other words, the screw could not easily work it's way into the wood so you pushed it in while turning it but not enough to have it penetrate all the way in. What you should use is a lag screw. I can't see what you want to do so I am going by what you are verbalizing. A 3 inch lag screw could be installed where the step has warped. You need a ratchet to install the lag screw. You can get these anywhere. Pre-drill smaller than the lag screw (1/4 inch bit would be fine) then place the screw into the hole and work your ratchet as the screw bites into the stringer under the step.
This will do either one of two things. It could split the step at which point at least you can replace it and see where to screw in the new one or it can result in the height of the step lowering enough for it to clear the gate. Cut the stripped screw with a hacksaw or just bend it back and forth with pliers or a hammer until it snaps off. You can also pull it out with pliers or the claw side of a hammer. Use a nail punch to push the rest below flush if you must leave some in the wood and fill it with exterior wood putty. You are correct when you say sanding it takes away the woods strength. You can't remove a warp by sanding it anyway.
Hope I Helped :)