How so?

Well in reality stay at home parents are engaged in unpaid employment that is completely unrecognized (or recognized as Not in Labor Force).

Unrecognized by who? Unpaid? You are investing in the most valuable thing that exists. Is somebody else supposed to pay you to raise your own children?

Wouldn't be a terrible idea given the long-term benefit to society at large of having young, healthy, educated adults. A relative shortage of them is a serious concern for a number of societies around the world.

"Society at large" is more hostile to young, healthy, educated adults than ever. The first step is to reduce the leeching of these people, so that they can have families as is the natural order.

There is increasingly an expectation that mothers will have a paying job. My wife is a stay-at-home mom and gets some strange reactions when sharing this (I just say it's way harder than my job).

Those people do not really seem mentally sound, and could be best ignored.

Lot's of people (like more than half the population we interact with) assume that my wife has a job, even knowing that we have four kids. I'm not sure your advice about ignoring all of them is sound...

>in society’s eyes

That’s all that really matters for many people.