Four strokes would work, they worked on the Indian mopeds. It's true that a two stroke creates more power than a four stroke, but the two stroke engines on mopeds are restricted to 2 hp or less, when a 50cc engine is capable of, from what I've heard, at least 12 horsepower. So a four stroke with equal restrictions would only produce half the power of a two stroke. Plus, four strokes have more low end torque and a broader power band, so different gearing could make them perform just as good if not better than a two stroke. They'd be more reliable, but when something does go wrong, it's a pain in the ass to fix because you have to mess with valves, cams, and stuff like that.
but I love the two stroke engines. They're so easy to work on plus, when tuned right, they produce amazing power. they also sound a lot cooler in my opinion. when you compare a two stroke dirt bike to a four stroke dirt bike, the two stroke sounds like it wants to kill something by viciously tearing it apart :)