Going to 15 in the example seems unnecessary to me.
A formal application of this method would probably break down on power of 10, as most people have a pretty strong grasp on addition and subtraction with single digit numbers. So:
4622 - 1873
7 + 20 + 100 + 2000 + 600 + 20 + 2 or 2749.
One nice thing with this method is you're only adding two single digit numbers for each digit of the final answer. So it's 2, then 6 + 1, then 2 + 2, then finally 2 + 7. My example turned out to be a little extra easy because it didn't have any carries in the final addition, but that's because I chose an example that explicitly looks like something you would get 1,000 times as a fourth grader and would be annoying to do in the traditional system.