The complaints are so idiotic because most of the complainers mechanistically apply the exact same principles without even realizing that they do so. Explaining all the steps is not convoluted, it is just explicit.
Take the example in picture 1. You would probably do the following:
Adding 9 and 7 first, you apply the fact that 29 = 20+9 and 17=10+7. Same as the vilified common core example. This gives you 16=10+6, and you write 6 in the result and 1 in the carry.
The 1 in the carry is only symbolically a 1, it's value is 10¹ instead of 10⁰. Writing the one in the carry in the 10¹ position, you apply the fact that 16=10+6, doing what the example to the right does, except you spare yourself writing the 0, that is implied.
Or maybe you don't do that and just mechanistically do what you were trained to do and apply useful properties of positional notation systems without realizing.
I assume (though I would need the context of the lessons to be certain), the point in the engineer example is to subtract the subtrahend piece-wise, starting from the lowest exponent, which is 10⁰. Jack starts with 10², which is where he goes wrong.
In my state, the same kind of complaints were voiced when set theory was introduced in math classes and when the synthetic-analytic method of teaching reading was introduced. Angry, panicky parents at parent-teacher-conferences
>My child has been to school for half a year! He can not read at all! He only learned words by heart!
Would be much better if he had only learned letters by heart, right?
Parents usually want their children to be taught the same way they were taught, and they don't want to hear that there might be a different way of doing things, maybe even a better way, because THAT IZ HOW YOU DO IT and I'll apply the breaks were my father applied the breaks, even if we are going uphill!