Yes, it does. Just did a FP there last week, in fact -- though I have also been there once or twice when the machines aren't open (i.e. on less crowded days).
They were up & running in early March also (MID WEEK). A couple of comments:
We were very glad we nabbed FPs (when we arrived at 5pm) since the standby wait was 45 minutes (at 7:30pm) and our FP wait was less than 5 min.
The FP machines are located conveniently away from the ride & near the main pathway in Frontierland....saving you 20-30 yards if you're gonna grab a quick FP (which i did while my family waited at the Jungle Cruise) for the family.
I actually would wait 30 minutes for BTMRR because the line moves so fast. If I were to stand in line for Autopia for 30 minutes it would seem longer than the 30 minutes waiting for BTMRR because of how slow it moves. I hope that makes sense.
We were there mid-week in February as well. I wouldn't wait 30 minutes for very many rides. Using TGM at Disney World has spoiled me, I guess. But why not use a FP? There aren't that many Fastpass enabled rides.
Thuner Mountain is one of those rides that can load pretty fast and the line moves really really fast. 30 min is really less considered that th rest of the park would have 60+ mins.