Generally the "even engineer dads can't make heads nor tails of it" objection is that the engineer dads didn't spend a couple minutes reading the helpfully coloured highlight box in the textbook. There has been a push in math to develop teaching methods that emphasize understanding rather than memorization. Thus 5x3 becomes 5x5x5 or 3x3x3x3x3 instead of "STFU and memorize your times tables."
A better example, also from Internet memes, is a procedure where you add or multiply a pair of larger numbers by breaking them down into component problems. 37 + 55 becomes (30 + 50) + (7 + 5) and some "parent" on Reddit or Facebook with add a comment like "why can't they just do addition like we learned??" Someone sensible will usually point out that people who are good at arithmetic will often use decomposition on harder problems if they're doing them in their head.
The teaching algorithms are pedagogical tools used to increase understanding or illustrate problems from different perspectives, not the final here's-the-algorithm-you-should-always-use".
I said that the 5x3 answer being marked wrong was likely due to a poorly educated teacher. No, primary school children probably won't be multiplying anything non-commutative soon. That was a joke. However, it is important not to instill, and then spend years reinforcing, incorrect facts. You shouldn't tell students things like "multiplication is defined as commutative" because that kind of thing will eventually screw someone up.