We've experienced two polar vortexes in Chicago since our pond was built in 2012. I'd say that qualifies as a cold snap. The ice was no thicker than it was in a "normal" winter here. We do keep our pond running all winter which probably has some impact on that, but the idea that ice will just keep freezing deeper and deeper when it suddenly gets cold - or stays cold over a prolonged period - has never proven to be true in our pond. Some years the ice is thicker, but it's never gotten more than four inches thick in even the coldest year. I've heard that having an ice cap can actually help keep the water warmer under the ice than if it's open water. Same is true with snow cover.
And the frost line in my area is 36 inches deep, for reference.
Lisa; I'm in agreement with most of your post, but will note that it's not a 'cold snap' that causes thick ice; it's prolonged periods of sub-freezing temps. Ice builds slowly after initial set up because now the water below is being insulted from the sub freezing air. The longer this goes on unabated, including cloud-cover with no sun, the thicker it gets. Our frost level here where I live in SE Mich is 42". I've played pond hockey my whole life and can tell you here, we can get 14" of ice easy, during the above mentioned times. That said, I don't think in the years prior to my winter pond tent, I ever got more than 10" and that's a guess during early spring.
You're correct re insulation of ice and snow, though there is a downside; with less light getting through, the algae below dies and this adds to the toxic gas build, especially without any breakage of the surface during these same thick snow/ice days. There is some benefit to actually shoveling off a portion of an iced up fish pond, just to let the light penetrate.
And I think most ponds/lakes require at least 4" to hold a person safely ( 8" - 12" for a vehicle, but I forget exactly).
I think the idea here though is deeper IS better in many ways, not just winterizing. And without at least 10' - 13' of depth, the water temp in a shallow pond will be the same as a deep one, all else being equal. A deeper pond will give more of the earth's natural temp (45-50 F) to the water and help keep the temp warmer. A shallow one won't get that benefit, so there is value to 'deeper is better' even if succes CAN be had with shallower.
Just some thoughts.