Those are not the assumptions, but there are indeed a great many problems with measuring productivity.

Usually, you only count work for money. Cooking dinner at home does not go into the statistic. Ordering dinner from a restaurant does. I would say that it is a problem that the “production” of leisure time is not counted. Of course, it’s not clear how this could be reasonably done.

“Productivity” already goes some way towards addressing such problems. It is usually GDP divided by hours worked (for money). US Americans work far more hours than their European counter-parts, so that their average incomes are much higher. Whether they are actually richer, depends on the value of “free” time. “Free” in quotes because it does not include necessary work like housework or healthcare visits.

If you look at a list of countries by productivity, you will find that it more or less matches common intuitions about what the rich countries are. That’s where people want to migrate to, so it does tell you something.

“Productivity” is how much a worker produces in an hour. Lower productivity means either that a people have to work longer hours, or make do with less. So, who cares? Pretty much everyone.