phase 3: I think that 'more popular' leads into 'better' because it also attracts more applications for open teaching positions and thus the school is able to choose the best teachers.
on one hand, more popular = better, because you're getting the creme de la creme selections of students who are top tier, bright and hardworking. spurred on by peers, kids may perfrom better. also comes hand in hand with proactive parents who 'can be bothered' and will contribute and do pull their weight (volunteer, donate, make time to input and help with school activities if neccesary etc)
on the flip side, more popular (like the poster after you said) = elitist.
with students and parents often very (read: too) competitive. teachers may be better, because they're under great pressure from parents to perform. but this push also trickles down to their students - some, who though bright may not thrive too well under such rigorous situations.
also, if i am not mistaken, the better schools get rewarded with more opportunities (better funding, enrichment etc)