I have small children I also make a pretty respectable Software Engineering salary, my kids have no real understanding of that. They just know Daddy works and has a job they don't really understand the finances, and seem just as content in the home we purchased recently as they did when we were living in an apartment, for them the realities of finances don't have major impact in their day to day lives.
In contrast my mother works in a Title 1 Elementary school[2], she spends much of her day dealing with children that are in chronically bad situations, it has been her observation that the biggest problem is that no one cares about the kids, not in a the parents hate them sense, but there is no one actively concerned with and thinking about and preparing for the child's future, there are all sorts of resources the school makes available to try and help these kids but it doesn't matter much because no one is invested.
When you have only one parent that one parent has is the sole person responsible for helping prepare that child, investing in that child, and guiding the child, and that kind of work is exhausting. It can be done my own mother is evidence of that, but when you have two parents now you don't have to be the point man on it all day every day you can share that burden with someone else, you have someone to help bear the burden when it is too much for you, and that shares the load.
That I think is an important part of what having a two parent household rather than a one parent household brings, even if finances aren't great it still means there is someone that can invest in the child.
But what doesn't get talked about with family (especially small kids) is that you're with them for 16 hours a day.
Emotions get messy. Someone is always upset about something. And the larger the family, the more probable that someone is upset at any given moment.
The nice thing about a two parent household is that (hopefully) one parent keeps a cool head while the other me be annoyed, etc.
It's so much of a help to have a partner that says "why don't you take five minutes" or "I think you were too hard on ..." And I can reciprocate the same support.
IMO this is a big factor but it really depends on the absolute value. Two low-income earners is not going to make a big difference, but two middle- or high-income workers is going to be a substantial improvement, although they still need to be 1) present and 2) dedicated.
Or, more importantly, selection bias. When people form a family, they eventually start having some disagreements. Those who have the skill to resolve them, de-escalate situations and find compromises, stay together. The ones who don't have these skills get divorced.
Seems like an easy thing to test, just adjust for income in single parents and compare outcomes. Easy enough that it's likely been done before and didn't materially change the outcomes of this kind of analysis