The counterpoint is that those teenagers suck because they missed out on years of practice because they were never given the opportunity to actually teach anyone anything meaningful before.
You are also presuming a high-stakes situation where mistakes are costly and go uncorrected forever. That is not realistic. Well-educated teachers are still vital, and a lesson gone bad presents even more teaching opportunities than a lesson gone well.
If an 8-year-old is helping a 6-year-old with math, then they are both learning, and the teacher is spending a lot less time watching either of them.
I think what we're seeing here is a general problem with outsourcing: you try to come up with a clean separation of responsibilities, but you end up with a costly mistake instead. In this case, thinking of all children as customers that are served by your employees, it would make a lot more sense to make it a collaborative effort between expert caregivers/teachers and students of various ages.
Offtopic, I can't help but wonder how much the tutoring reinforces what they were probably certain to forget anyway. In the same way that blogging about a subject reinforces your understanding of it (e.g., if you can't explain it, you don't know it), I suspect that tutoring a younger generation would help those tutors cement that knowledge more deeply into their minds than just retaining it long enough to pass whatever tests they're given in the short term.
At the end of the day though, your dojo analogy is most precise -- senseis don't just promote every student to assistant instructor. They carefully curate based on knowledge, discipline and behavior. Having every older child tutor the younger is probably unnecessary, and would likely lead to chaos, but if teachers chose only those students capable of helping out the younger children, even assuming some failure rate in selection, it would probably be a dramatic improvement over the current situation.
I've learned eg cycling from practical demonstrations of my slightly older cousins.