Sure, adoption markets basically already exist, why not make them legal?

Not only are wealthier people better candidates on average because they can provide for the material needs much better and will on average have a more suitable psychological profile (we can impose legal screening of adopters too, so they need to match other current criteria before they can legally buy on the adoption market if you feel uncomfortable with "anyone can buy"). It also provides incentives for people with desirable traits to breed, far more than just subsidising them having kids of their own.

I think I may not have gotten the point I was trying to make across - I don't think all harm is of the form "violation of property rights", I think the reason "property rights" are a thing we care about is because their violation is harmful.

Would you permit a market in infants?

An interesting question, but not one I've thought about. If what I've said above is tells you what you want to know, I'm not going to try discussing this here. Otherwise I will.

One of the standard topics in economic approaches to the law is to discuss the massive market failures caused by not permitting markets in infants; see for example, Landes and Richard Posner's "The Economics of the Baby Shortage". I thought their analysis pretty convincing.