Today, I discovered CallContext, and it is good. I will probably use it lots instead of being tied to HttpContext. Sure, there's a downside in that we won't be able to switch threads freely, but hey, at least it works in non-Web environments, so I don't have to reference System.Web.dll from all my core and application logic classes.
2 days later: I have torn out most of the remaining hairs from my scalp. It turns out that ASP.NET doesn't - in fact - guarantee that your entire request will be executed on the same thread (see here for more details). At least the HttpContext class still works; even though it depends on an underlying CallContext (when an HttpApplicationFactory initialises an HttpApplication, the new HttpContext ...blahblahblah I'm tired) the HttpContext is moved from thread to thread whenever one of this switches occurs.
What I have learned: don't trust simple Google searches that tell you each ASP.NET request will only run on a single thread.