Is the bottom line on this one that I just can't use a polarizer when the wide angle is so much?
Exactly.
How do other landscape photographers handle this?
You can often read, that there is no substitute for the polar filter. This is true regarding its function of eliminating reflections, but simply the darkening can often be substituted by Photoshop.
I am selecting the sky and applying a curve, which is good for the contrast on clouds as well. Sometimes I change the color as well; in other cases I develop a different version from the same raw file, adjusted for the sky. This can be a very tedious process, and if there are fine structures, like thin, dense branches, leaves, then it does not work.
Example if I had developed the raw for the sky, then the rest would have become too dark.
I admit even to planting some "stock sky" sometimes, but that's not easy; the planted sky has to closely resemble the original at the intersections with the background. Moreover, it can cause an impression of "unnaturel" (what a surprize), for example because of incorrect shading. In one case I was looking at the result for a while; I did not like it, but I did not know why. Finally, I found it: the clouds' reflections in a small pod contradicted the planted sky.