I suspect that most teachers have excellent social skills; that has been my experience. How could they not have? To teach well you have to be able to understand and communicate with a wide range of character types, and you have to be confident enough to speak to and influence a whole room full of other people. Teachers see things (like everyone does) primarily from their own experience, as good communicators.
But lots of people don't have good social skills. Sometimes that is because of upbringing and school experience, but a lot of the time it is innate. There is plenty of research showing that introverts and shy people are born, not made. No amount of informal learning will teach those people to have really great social skills.
I cannot imagine anything more awful for my DS than to have to undergo extended informal learning. Just getting through the informality of reception nearly destroyed him. That year was the worst year of my life. All these people who want children to do maths verbally (by which you presumably mean orally), can you imagine being a shy child who is afraid to speak to the teacher, but who isn't given the opportunity to show what they can do through writing rather than speaking?
Someone mentioned earlier about children who can decode things they cannot comprehend. My DS was accused of this; he was actually just afraid to express himself orally. Asides from that, decoding is a really important skill that has many uses other than in literacy. DS still loves code breaking and is giving lots of opportunity to do it at secondary school.
Many children come from backgrounds where they don't hear adults using a wide vocabulary; the only way they can learn to comprehend more is if they can decode more first, because they can only learn that wider vocabulary through books. A child has access to a teacher for a few hours a day, but if you teach them to decode the child has control of knowledge and choice over what they want to learn.
The only study done in this country that looked at the outcomes of informal learning in terms of social inequality showed that, in comparison to traditional methods, the play based curriculum widened social inequality. This was reported in the TES and then ignored by many teachers.
My DD had a great time learning through play, but she had great social skills.
Children are individuals; they don't all learn in the same way, or benefit from the same methods.