When we were just having those stale-cache issues, with redirected pages and the like, it seemed that you got a consistently stale page, not the randomly stale page you'd expect if each refresh got you a different squid server with a different stale version of the page. Moreover, the effect seemed to be browser-dependent: you could open two different browsers on the same redirected page, refresh each browser several times, and the two browsers would consistently get the same two *different* consistently stale pages.
Do the load balancers in WMF's server farms attempt to associate sessions with servers? At first I thought this unlikely, but I guess it could significantly improve the hit rate.