I cut you choose is not fair. I actually heard an example in real life about a week ago when something was being cut for two people (they weren't following the "I-cut-you-choose" protocol, just trying to be fair). The person doing the cutting was literally saying "It is very hard to cut it so they're both the same."

This is true. Many things don't naturally cut into two equal pieces despite the person's best attempts, and in certain cases it may be impossible. (For example, if there is are "valuable" toppings that are indivisible.)

It is just plain easier to choose than to cut. The worst case choice for the person choosing is that they are indifferent to which piece they get, but the worst case for the person cutting is that they get a worse piece.

Choosing is strictly always better than cutting.