Wow. Thanks for that, Konrad. I would never have expected that. That's a strange limitation. I hope that isn't the case for the watch I just bought.
After a lot of reading, I decided to gamble. I bought a cheap Pyle alti/baro + thermometer watch today for $35. It has all of the functions of a $500 Suunto and, in addition, it records temperature minima and maxima.
It may stop working the day it arrives, but I couldn't identify any more promising option given my criteria. I'd like something that I feel more confident in, but I had to rule all of the other options out.
A Casio would be ideal. I have gathered that they are by far the most waterproof alti/baro/thermo watches on the market, they have very long battery lives, and malfunctions seem to be very uncommon. But none of the Casio alti/baro watches can record temperature. None of the Suunto watches can, either (and there are a large number of Suunto malfunction stories, anyway).
The HighGear, Silva Tech4O, La Crosse, and Timex watches have water resistance problems that seem embarassing. Most of them are only rated to 30 meters (some as little as 10 meters), and they sometimes fail to achieve that. There are plenty of stories (particularly about the Tech 4O's) about water inside the screen of a new watch after wearing it for a few minutes in a drizzle, then customer service reps explaining that they don't recommend allowing the watch to get wet. The Casio watches are all rated to at least 100 meters and I have yet to read about leakage.
The Pyle PAW1 has low water resistance, but it can't be worse than a HighGear or Tech4O. It has better claimed battery life than any Suunto, and all of the same functions. I'll post updates once I've used it a bit.