Good news: I was right about transformer winding selection. I switched to the heavy gauge secondary winding, and all is well. I can now crank the thing up as high as my household circuit breaker allows.
Bad news: As I feared might happen, the cutting wire became brittle at high temperature and snapped during my first test at a higher voltage!
In anticipation of this problem, I had prepared a quarter inch copper tube with a length of nichrome wire running through it, electrically insulated by a high temperature fiberglass sleeve. The idea, suggested by Lars, is that the tube, not the wire, bears the brunt of the ice pushing against it (stainless steel would be ideal, but copper was fine for testing). This certainly allowed the wire to get very hot; I could see it glowing brightly right through the fiberglass sleeve! It even left a couple of black marks on the sleeve. Unfortunately, it didn't produce enough heat to cut at a useful speed due to the wide swath of ice that must be melted for the tube to pass through. Maybe a narrower tube could be used, but I haven't found a smaller high temperature insulator to keep the current running through the wire and not the tube.
Then I tried a new length of nichrome in the original setup to see how it works at lower voltage. I adjusted the tension somewhat lower too. I was able to cut at a reasonable speed (at least two inches per minute) with 14V across eight inches of wire, but I had another breakage when I turned it up to 16V. It cuts more slowly than I had hoped, but it is certainly usable.
Two inches per minute may sound quite slow, but I believe it is faster than the boiler we used last year with some success. The benefit of this technique is the ability to make precise, intricate cuts that are perfectly perpendicular (or at some consistent angle) to a surface of the workpiece. For many mechanical parts this benefit could outweigh the slow cutting speed.
Next I would like to try a lower gauge cutting wire which should fail at a higher power level. My guess at this point is that the tension doesn't matter much (the second break happened under very little tension) but that the breakage occurs due to the wire becoming so brittle at high temperature that very little physical stress causes it to break. The failures tend to be located at the hottest part of the wire, between the ice and the alligator clips that deliver current. I'd also like to try to minimize that gap, but it can't be completely eliminated.