This is not a view shared by developmental psychologists. Playing with children their own age is fertile ground to learn how to share, how to take turns, etc. As far s the child is concerned, they are on even footing in the social hierarchy with their peers and must resolve conflicts with their peers without falling back on defaults. Moving into adolescence things become more complicated and nuanced-- coping with peer pressure, dating, developing a sense of self etc. But that's all very critical.

Do groups of closely-aged children (or people in general) act as peers, or do they cue on other factors to form a hierarchy from? This is what the LotF effect is. In the absence of an established social hierarchy children will create their own based on arbitrary things such as body size, clothing, manner of speaking, etc. In schools we see this manifested as bullying.

Having a child interact outside their peer group gives them a template of maturity that they can emulate within other children their same age. The established hierarchy of age may make it less likely that they will resort to arbitrary hierarchies and could be measured by a decrease in bullying between children of the same age. (There may be age related bullying as older children abuse their seniority.) They will still have to learn the things you mention, but instead of stumbling across the rules on their own, or having them dictated by the fiat of an authority figure, they learn by the example of those who came before.

Then as those children mature they are encouraged to work with younger children and act as role models. When seeing they can correct the mistakes of the children following them, it provides positive reinforcement of the good behavior.

I was curious about this, so I searched for developmental psychology on the difference between mixed-age and same-age socialization. But what I found (from a very brief search) didn't match what you said. An example study is this - Children’s Social Behavior in Relation to Participation in Mixed-Age or Same-Age Classrooms:

"Findings suggested a significant positive effect on children’s prosocial behavior as a result of participation in a mixed-age classroom context. In addition, fewer children appeared to experience social isolation in mixed-age classrooms than in same-age classrooms. Aggressive behaviors were significantly less likely to be noted by teachers in mixed-age than in same-age classrooms." - http://ecrp.uiuc.edu/v1n1/mcclellan.html

But that's just one study. Is it an area of ongoing debate in development psychology? What sources would you recommend for the other side?

What you're looking for is a study that shows something like the following: The difference in social development between at least 2 groups of children: Where one group did not socialize at all, and another that socialized with similar-aged peers.

Also, while the study you link is interesting, it's really just looking at pro and anti-social behavior over what seems to be one school year. It's not doing a formal assessment of social development over a long period of time. There's certainly evidence it could be beneficial long-term, but long term effects aren't studied in that paper.

I think your mistake is that this does not describe homeschooled children at all. They just socialize in locations other than prison-style government run daycare centers.

If RMS were a homeschooling advocate, this is the kind of statement I'd expect from him.

If you are on the "right" side, there's no need to exaggerate.

In what way is that statement an exaggeration?

Do I need to spell out how public schools are run like prisons? Or do you not believe in their critical function as daycare facilities?