Did you really mean the ENTIRE stick?

The ball shape in the lever's base is held in place by a bracket which should prevent any vertical movement while allowing movement in the horizontal plane. Even if somehow the selector rod had come off the lever, you should still not be able to move it up or down. In fact, here's a photo on which the selector rod is missing:

However, if the snap ring had come off, you would be able to lift the lever out of the bracket, but I don't see how you could push it down...
In addition to inspecting from the bottom I would also lift the rubber dust boot and inspect from the top.
All this does not seem to explain where the burning smell comes from, though. Be prepared to deal with more than one issue.
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