I went and tried the grip tape, and does indeed chew up the tires. But it also works very well! I did a few sprint intervals, and the tire afterwards was smooth, warm and sticky. It seemed to clean off the top layer of rubber. Then, after some standing starts (only three from an enduro), I found the tire had striations and some granules from the grip tape. The tape itself had actually started to wear all the way through!
At this point, I have seven spare road tires to use, and only a few feet of grip tape. I'm more concerned about how much tape I'll need, rather than tires!