The modulus of elasticity of titanium is 16 million psi, and for steel it's 29 million psi.
The main mass item for shimmy or vibration is at the bottom end of the leg, so the density of the leg is only a small contributor to the natural frequency of any vibration.
So for the approximate change in frequency of the vibration would be the square root of 16/29 or about .74. The new titanium gear leg would vibrate at about 74% the frequency of the original steel leg.
One of the easiest ways to reduce vibration is to increase the natural frequency of the member that's vibrating. This is usually easier than changing the forcing function, which in this case is the rotating wheel and tire and the effects of possibly scrubbing on the ground.
Another way is to add damping to the system. The wooden dampers do both and are modestly effective.
You certainly don't want to lower the natural frequency of the gear leg.
Bottom line - if you used titanium gear legs, assuming you could make them strong enough, they'd be more likely to vibrate.
Now the caveat - There's nothing preventing a person from developing a titanium gear leg that would work. It would have a larger diameter than the steel and would be a totally new design, and would require a new mount.
Dave