... At high noon, the ring sits at the horizon, 360º around you...
In this situation, it is possible to use a polarizer on even the widest lens and get even polarization of the sky from edge to edge. The polarizer nicely cuts through the haze near the horizon, while the sky higher up is less affected by polarization but is a deeper blue anyway, so the effect looks even top to bottom. I regularly used a polarizer on my 20mm lens (and I'd use it on my fisheye if I could!)
Even when the sun is lower and the band of maximum polarization arcs up into the sky opposite, the polarizer can still be used effectively with some care, depending on how you frame the sky, and whether you adjust the polarizer so the sky is fully or only partially polarized. It's only when you get a dark band of polarized sky running down the middle of the image that it doesn't really work. So I don't believe it when people say that you can't use a polarizer on wide lenses, you have to judge each scene on its merits.
The effect is the same with film and digital, but in my experience, polarizers results in a strange loss of contrast on digital that I never found on film, even on the best multi-coated filters. I suspect digital sensors are more reflective than film, so reflections from the polarizing foil itself (sandwiched between the glass) may be the cause.