If you were to bring a bike, you'd need to dodge people left and right during the spring, summer and fall on both the incredibly steep climb up and the crowded descent. You'd also be the person disrupting the traffic flow. I'd leave the bike home.
When I last hiked the trail, someone had a mountain bike. I'm not sure how far up he went as he came down as we were still heading up. Even riding up to the hotel would be worthwhile. The ruins are great.
The trail as far as the hotel ruins (1.9 mi.) is a dirt road that's passable by trucks that go up to the antenna mast. I'd have to say it's really too steep to climb on a mountain bike and much too crowded for downhill riding. I don't think bikes are allowed but I'm not sure. Beyond the hotel, definitely not. It's a narrow hiking trail, not single-track.
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