This strikes me as being a somewhat obvious difference, but a (tethered) ice diving cert is pretty quick and dirty with not a lot of academics, in water training or skills with only a couple of dives that tend to be comparatively short, no consideration for or coverage of "lights out" exits, lost line drills, gas sharing, etc.
Even a cavern class is two full days and 4 dives minimum, Intro Cave divers will have 4 days and 8 dives and Full Cave adds another 4 days and a minimum of 8 more dives, and those dives all tend to be at least an hour long and are drill intensive.
So it's maybe 40-60 minutes in the water mostly looking around to get an ice diving cert while it is 4 to 16+ hours of in water training depending on your cave cert, and that does not even touch the in water experience that may be gained between cave certs.
From that perspective, unless you seriously plan to ramp up ice diving training in terms of both duration and intensity, we are not even remotely considering teaching cave styl ice diving. The basic skills are just not there for the average ice diver candidate.
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Personally, I prefer non-tethered ice diving, but it is true that their are situations where that is a bad idea. Back in the day we used to do drift dives in the winter below a hydr electric damn. The current and turbulence used to keep the ice off that first 1/2 to 3/4 of amile or so before the river was completely iced over. When you did that, you needed to be very aware of where you were. With some experience you could "read" where you were by changes in the bottom as you got farther donwstream, but you also needed to be prudent enough to make sure you were close to the exit side of the river and exiting well before the river froze back over. It was something a lot of us did, but none of us recommended it to new divers or divers with limited skills or limited SA. Not everyone was that thoughtful or that prudent in their diving and we had a deputy drift under the ice to be recovered later that spring several miles downstream.
Similarly, I have been on recovery dives under the ice where a tether was still the way to go due to zero viz - often rooting around in 3-4 feet of very soupy silt on the bottom of a reservoir.
Places where non tethered ice diving include lakes or quarries with decent depth nad good viz. One of my favorite ice diving activities is on larger lakes where only the bays and cove may be frozen over. You enter on a point, tie the lie off in open water then penetrate back under the ice. You also have compass heading for back up, but a line is a sure thing.
I do agree with the sentiment to avoid entering a lake through a gap in the ice along the shore line - any shift in the wind and that gap can dissapear.
