There might be companies that would like to do this. It sounds like kidnapping, which is illegal in almost every country. You would definitely need government approval or support.
So many weird things in one paragraph. I saw others address the family and child care aspects but still I wanted to reply. Who is "we" in "if we want to exploit"? I very much doubt that "we" is a significant number of people. Denying a working couple a child doesn't increase the workforce, you have to create jobs for someone else to increase the workforce, at best this is a way to stop temporary losses not increase it. But, even then usually someone else is hired for maternity/paternity leave positions in places that have it, or replace them in places that don't. Next, what exactly do you mean by "society being as productive as possible"? The classic example of how to reduce unemployment is to enslave all the unemployed and congrats you have zero unemployment. The point of that example is to show that the means used matter. The same could be said for your "society being as productive as possible", just force every person, including children to work as soon as it's physically possible and congrats the world is much more productive than now. Any 3 year old could pack boxes of light stuff at an Amazon fulfilment center. I don't think a lot of us would like to live in that world. Also, exploiting more of the workforce doesn't connect directly to society being as productive as possible. For example you could cut work weeks to 20 hours a week max, that means almost every position now would need a 2nd person, so in theory the workforce would double, but productivity doesn't increase much if at all. I'll say the same thing I said in another thread a while back. Plenty of conservative people don't want their wives in the workforce. So beyond all the points made by others above about family and children, a significant percentage of humanity will resist any plan that tries to get them in it.