At least in sailing, you can go a lot faster than the wind; I assume the same is true for these blades.

it would be except that the goal of a wind turbine is to generate power rather than go fast

How do you go fast except by generating power? Force times distance over time, that’s how sailboats move too.

It's not the blades that have to go fast, but the generator.

More specifically, power is Torque x Omega (rotational speed) for a rotational system

So yes, faster means more power. Torque is limited by the generator and how much power is "being pulled" from it. (which is a bad description but there you go)

How can you travel faster than the wind that is pushing you?

Apparent wind, and angle

Even if that where true (note that the wind is not pushing the blade directly away, but sideways), the tip of the blade is not responsible for most of the force (since it mostly depends on the cross section area, intuitively).

In other words, wind pushing the middle of the blade could accelerate the tip, even if the tip only produces drag.

Not necessarily!

https://www.youtube.com/watch?v=yCsgoLc_fzI