How do I use a token to set the default value of a date field to the current date?
Leave the "set default value" button alone. Something about it makes it seem like a really good button to click for setting a default value, but you don't need it. Just enter the token in the default value input on the advanced tab and click save.

This seems simple enough in hindsight, and I'm not sure why it took so long for it to occur to me to try it. Hopefully if someone else has the same problem, this answer can help save them a few minutes, or perhaps even a longer period of time we won't mention here.