Illich wasn't against school per-se. He was against compulsory school, centralized pedagogy, and other attributes of a "modern" (ala 1970s) schooling system in Western social democracies.
He was particularly concerned with how the tenants of modernism affected developing nations (where he lived for much of his life) by pushing traditional methods into forced obsolescence in favor of ever-increasing reliance on social institutions.
His views on policy may or may not be on point (he seems most comfortable in the area of principle and theory), but the conceptual core that relates his views on education to his views on cars and medicine merits consideration.
He similarly wasn't against scientific medicine, but he thought the medical bureaucracy had outgrown its utility: that advances in specialized medicine were too expensive and served to prop up artificially unhealthy lives.
A strong line of his argument is that past a tipping point of size/power, institutions will create the needs they serve. We create these artificial environments where it's hard for most people to thrive and then we create social institutions to compensate: a social prosthetic.
The result in Illich's view is increasing passivity and helplessness: a petulant society composed of a disempowered citizenry.
The argument no-doubt merits a balancing counter-argument, but I find it compelling and it goes deeper than a simplistic dismissal of school.