Having just received a new transformer in the mail, I was able to piece together a first attempt at a new power supply tonight. This allowed me to properly test the hot wire cutter for the first time. I am happy to report that it sliced through a block of ice quite easily. I had it up to about half of (theoretical) maximum power, and it cut as fast or faster than any hot wire I have used to date. This was the first time I have ever seen 19 gauge nichrome get red hot. I was afraid that it might become too brittle when hot and snap under the tension, but it remained intact. Unfortunately I had to cut my test short because the new transformer started making ominous crackling noises and then started to smoke a little bit!
The unit is rated for 2000 Watts, but, in hindsight, I think that rating is only for one of its secondary windings. It has a primary winding of heavy gauge wire rated at 100V, a secondary winding of the same wire at 95V, and various lower voltage secondaries from 15V to 30V. I plugged in a variac with its variable tap connected to the new transformer's primary winding and attempted to use one of the lower voltage taps to power the hot wire at 0-30V. It started to crackle at about one quarter power and smoke at half.
I think my problem was that the winding I used does not have thick enough wire for my power level. Instead of using the new transformer to step down from high to low voltage, I'll just use a lower voltage tap on the variac and use the 95/100 tap on the new transformer for isolation.