Take a better example, like:
Doing this in your head the traditional way would be hard.
Not really; the steps are (working from the right, of course):
1 < 8, so 1 becomes 11, 2 becomes 1, and 11 - 8 =3
1 (previously 2) < 4, so 1 becomes 11 and 3 becomes 2, 11 - 4 = 7
2 (previously 3) > 1, so 2 - 1 = 1
Answer: 173. Took me all of 10 seconds. I needed to remember at most 3 pieces of information at once (the fact that I borrowed plus what digits I had already solved). That's well under the 5 - 9 items that people can hold in short-term memory. With this method, I just need to know how to count to 20 really well, and if I'm really stuck, I can use my fingers + toes!. I use this method ALL THE TIME when tipping, to figure out "what tip to I need to make the bill X".
Granted, if I'm subtacting 10-digit numbers, the "traditional" method can get tough to do entirely in your head, which brings me to...
[T]his is the kind of math that makes people think they can't do it without assistance from paper or a calculator.
When was the last time you NEEDED to add/subtract a 3+ digit number and you didn't have a pen/paper or a calculator with you? I don't even have a smart phone, and I've got a calculator on my cell phone if things get really tricky.
But doing 52 + 21 is much easier, and doing 73 + 100 is also quite easy.
Where did 52 come from?? There's no 52 in the problem anywhere! And why are we adding 100?
The "traditional" method only looks at a single digit at a time, so you only need to know how to add 2 single digit numbers (and carry or borrow). With your method, you need to first know that 48 + X = 100, so X = 52. You're no longer doing arithmetic in your head, now you're doing algebra in your head!