"single parent households" are precisely why these levers are important: among other things, they help reduce the disadvantages some kids have due to being raised by an impoverished single parent, and gives those kids a leg up in a way which will foster more stable home life and less likelihood of themselves becoming single parents.

Not only that, but more resources and more stability help foster successful relationships. If you want more two-parent households, make it a lot easier to have and care for a child.

It can absolutely matter, and in fact it is all the more important in a single-parent household.

You’re right that single vs. two parent household is the largest contributing factor. You’re wrong that it means that no other factors matter at all.