While the proposed idea is very nice, I feel that one limitation would be the fact that children and young adolescents don't always have permission from their parents to wander around on their own, especially to go to talk to strangers without any supervision of knowledge from their parents.
This issue could be solved by giving the kids access to this kind of emotional support at school. The school is a controlled environment that the parents trust to be safe for the kid, but have no direct overview on.
For this to work, the emotional support system should be included on the school but completely independent of teachers or other staff of the school. Actually, in Spain and other countries, some schools are starting to employ psychologists, especially those schools in conflictive, high-complexity areas where children tend to have more psychological issues.
Can we discuss about what would be the best way to recruit and oversee these volunteers? How many hours a week should they dedicate? What guidelines should they be given? And should they be available during class hours or after class?
However, I believe such approach is not enough. I think that a single psychologist can't even begin to address all the potential issues. For this reason, I believe that the idea of having psychology and medicine students, who could be recruited as volunteers, and overseen by an expert, is amazing, since it would provide a great learning opportunity for the volunteers. Also, since the volunteers would be closer in age to them, this could allow the kids to talk about topics they wouldn't talk with "adults", like sexual topics, for example. There obviously should be some basic guidelines, but I believe this could help wih a lot of traumas and complexes that teenagers develop because they lack the proper support.
Additionally, the volunteers could organize activities with a reduced group of children, of whom they are responsible, in order to get closer and show them that they can also share their problems with each other to find support.