I can't tell from your comments whether you realized yet, but the trick is only useful if you add several 256 bit numbers. You can add up to 2^13 numbers (each one 256 bits) before you actually have to do the four carry shift-outs once. In effect, you do reduce the number of carry operations (to four, albeit slightly more involved), and that is what ends up saving time.

There is no performance gain if you only add two numbers, even though that's the setup the article discusses for 90% of its length. It seems that this is leading to some confusion.

You are reducing the number of carries. The article explains it clearly. If you're not convinced by the article, I won't try to convince you. (Edit: the last thing I'll add: you are reducing the number of carries for 64-bit additions. When adding 13 51-bit numbers, that does not count as carries instruction-wise, as it's still add, not adc.)