> One of the teachers demands to know who is picking each kid up if they have to stay after school and will not let the child leave the building otherwise.

The school needs to know who is taking the children out of it is helicopter parenting now?

>At that age (at that same school), I took myself to school and biked myself home if I stayed after. Hell, I was taking myself to my own medical/dental appointments at that age. 

One teacher can't dictate school policy. If biking is allowed, it is allowed regardless of what one teacher says.