Do data-dependent rotations have any advantage over fixed rotations? There are many ARX ciphers, however most use fixed rotations.

I know data-dependent compared to fixed rotations are:


patented (expired)
harder to implement constant time
more expensive in hardware
harder to analyse
attacker can control rotation to some degree


Do data-dependent rotations have any advantage over fixed rotations?
Compared to fixed rotations, data-dependent rotations improve resistance to differential and linear cryptanalysis. A fixed rotation has no effect (beyond helping with diffusion) in the probability of a (xor-)differential characteristic, whereas a data-dependent rotation also introduces differences in the rotation amounts, which brings probabilities down. There's a 2000 paper by Scott Contini that quantifies this advantage.

Successful attacks against ciphers using data-dependent rotation have focused on trying to avoid differences in the rotation amounts, or to fix them to some value; this was the case for RC5. Later designs like RC6 and MARS made this harder by using more complex functions as the input to the rotations. This is an interesting paper studying the effect of varying that complex function in RC6, and how it affects the overall security of the cipher.

That said, the disadvantages you listed outweigh the potential advantages, and data-dependent rotations are generally not employed in new designs.