Thanks @krisdrum for the thoughtful response. Sounds like we have similar issues. I grew up in Jersey, but now I live 2+ hours from any hockey shops, so the DIY help is much appreciated.
I'm surprised to hear that they shimmed the rear post of your skates only. I'd think that would put weird torsional stress on the holder, twisting it and eventually making the blade loose. Do you know why they didn't do the front too? Did this cause any structural issues?
I use the in-boot shims like you describe in my cycling shoes (and my ski boots) to great success, where like you said, the foot is pushing against a fixed flat surface; it brings my joints into alignment. But in my skates -- where the blade can pivot freely on the ground -- it feels like building up the medial side just rotates that outside edge further from the ice surface. Hence, the reason to shim the holder on the lateral (outside) half.
Regarding remounting blades vs shimming, maybe I need a better understanding. It seems like they combat two different parts of the problem. Shimming seems like it helps bring the blade more perpendicular to the ice, so you have equal ease of using either edge. Remounting seems like it would be the better way to truly locate the blade under the natural balance point. (As for why this isn't done as much in hockey, I assume it's because hockey players are curmudgeons who won't buy anything non-traditional, and skates that easily allow this won't sell; it's not a matter of it being ineffective.)