A great idea and you've demonstrated that it works! Divide and conquer.
I've always used my front end loader to pull smaller stumps. There are two advantages. First it a bit safer. I've seen too many chains slip or break. If that happens they often head right towards the person in the tractor seat. Second, you can get more torque from the bucket lift over a shorter distance. Just put the bucket in the dump position, attach the chain tightly, with no slack, then tilt the the bucket up. Admittedly sometime you just lift the back end of the tractor.
One reason this works is because trees don't grow roots to withstand an upward force. They do grow to withstand lateral forces. A side effect of this is that if a tree has been blown off center and allowed to continue to grow slanted it is VERY difficult to push over from the side with the larger, open angle. If you push the tree facing the narrow angle it actually goes over more easily.
I will try your cutting technique along with the front end loader. It should help with larger stumps.