Hope you know 1). the Ph.D school/program ranking does not mean all for its graduates
2). Just on average, better schools' Ph.D holders have higher achievement, but not for everyone
3). In US colleges, department chairman is not necessarily the best in academy, rather, most professors do not want to be a chairman or undertake "administrative" jobs which will cost their time on writing and publishing academic papers...
If one professor wants some students to work for him, he might take the chair position to abuse some power of admitting students. Or enjoy the process of giving comp exams.