All the mainstream 6th gen consoles had first party support for wired ethernet; although only the Xbox had it built in (well, the slim ps2 had it built in too). I'd expect they've all served web pages at one time or another.

5th generation could probably make it happen, but without ethernet, you're looking at a modem or serial interface, and in today's world that's almost certainly talking to something else in your house that's a better host. :P Wikipedia categorizes the Apple Pippin as 5th gen, and it's more or less a Mac with a weird pinout for the PCI slot, so I'd guess it's the easiest to get ethernet with; without resorting to something that has more smarts than the 'host'

Even with Ethernet, aren't you're plugging the console into a switch that can probably serve webpages as well as the console?

Most of my switches are unmanaged, so probably not. Maybe the dsl or cable modem or whatever can serve pages, but especially if it's isp owned, it might not be easy to.