Please re-read what I said. I asked what a nursery can give that a "hands-on, involved, curious, and loving parent" cannot.
A parent like that would take their child to play groups and classes and have play dates so the child will have plenty of socialisation and friendships in which they can explore social etiquette, social norms, manners, etc.
In addition, they would take their child on outings to the park, library, farm, woodland, etc which a nursery would not.
My own toddler goes (and will continue to go indefinitely) to groups where he mixes with people of all ages - from babies younger than him to older adults the same age and older than his grandparents and everyone in between. In terms of learning social skills, that's surely far preferable to be artificially placed into a room with only others born within 12mths of you?
It's unclear what age-bracket the OP is talking about but 2yr olds are not developmentally ready to share or "understand peer emotion" and throwing them into a group setting and expecting them to do so is just a recipe for absolute chaos. Ask anyone who has worked in a toddler room. 3&4yr olds don't have to be in a formal setting to learn these skills either.