The AoA of the fan blades varies throughout the speed range, very similar in principle to a fixed pitch propeller. However, the change in AoA also depends on the propulsive efficiency of the engine.
This variation in AoA is determined by the change in "down" wash at the fan blades (think of the fan blade like a wing). This downwash is for to two reasons: forward speed (TAS), and the inherent downwash (jet-wash) due to the production of lift (thrust) itself. On a turboprop engine with high propulsive efficiency, this latter downwash is low, but is relatively high on turbofans (and is even higher on turbojets). The net downwash as experienced by the fan blades is a sum of the two.
When a turbofan is operating on ground at zero TAS, the effective TAS at the fan blades is quite high. This is due to the aforementioned downwash at the fan blades due to the production of thrust itself. This downwash reduces the effective AoA of the blades enough to prevent it from stalling. As aircraft TAS increases, the net downwash increases, causing the effective AoA of the blades to decrease.
However, on a turboprop with a much higher propulsive efficiency, the initial downwash (due to thrust production) is relatively low, and it's influence on the effective AoA is much less. When the aircraft TAS increases through the same speed range as in the turbofan case, the percentage increase in the net downwash is much higher, and so is the change in AoA. The only way to maintain a reasonable AoA is through the use of variable pitch propellers.
In the turbofan case, the change in AoA as the speed increased was significant, but still low enough - a variable pitch mechanism would be able to provide much less improvement, meaning that it's use is not feasible.
If you find this difficult to accept, think of the extreme case: if the initial downwash was infinite (blade pitch would be 90°), an increase in TAS through any speed range would have no effect on the AoA, and so a variable pitch system will be absolutely useless in this case.