Apple certainly doesn't run any web technology—they don't have skin in the web-app game like Google and Microsoft and Amazon do. (They have a web store, but on iOS it heavily suggests its native-app equivalent.)
But that doesn't mean they want their users to not use the web. Web-browsing is a feature; Apple invested a lot of energy into WebKit for a reason.
Apple just don't give a damn about the supply side of the web; they don't care if they kill every web business in the process of giving users the best web-browsing experience possible.
I have high confidence that if Apple could integrate something like e.g. Tor into Mobile Safari without a latency hit, they'd jump at the chance. Better privacy for their users! Breaks analytics and ad targeting? Who cares?