That kind of caching is actually extremely rare in my experience. It's difficult to set up for all but the most simple static sites, and since it isn't available by default the vast majority of sites don't bother.

The times I've implemented full-page caching (generally using Varnish) I've set it up so nginx runs in front of Varnish handling SSL termination, which means that I still have SSL support even though I'm serving through a cache.