Why is variable rotation uncommon in cryptographic primitives?
One issue is that data-dependent rotations (such as you describe) is patented by RSA data security (or, at least, was, the patent may have expired).  RC5 and RC6 was created by the holder of this patent, however such a patent could be enforced against someone else, and so people have shied away from it.

More minor issues would include:


It is likely to take variable time on lower-end CPUs, and hence potentially vulnerable to time-based side channel attacks.
Such variable rotates are moderately expensive in hardware (when they need to be implemented in constant time); yes, you can construct a barrel shifter, but that takes a number of gates and interconnects.