Were these schools of thought truly as influential in the foundations of western society as he claimed, or did these fade out in place of more modern schools of thought as time went on?
Looks like you've got a tautology on your hands! Most people use "Western Civ" to mean "the legal, philosophical, and cultural traditions originating in the Classical world and continued by Christian Europe," so of course it's built on those: that's how it's defined!
As I've discussed here and others have here, saying Western Civ is based on one thing or another is a mostly vapid argument. "Western Civilization" isn't really a thing, in that it doesn't really exist outside of European and North American academics and politicians trying to prop it up as A Thing. Really, it's just a self-replicating meta-narrative, as described here by /u/agentdcf. This comment from /u/hillsonghoods gets in to the origin of the term "Judeo-Christian."