| Saying pitch doesn't affect thrust of a prop is equivalent to saying | AOA doesn't matter to lift of a wing.
That sounds clever and all, but It's not quite so simple.
Your standard prop is an airfoil, but it's not an airfoil with a constant angle of attack. Near the root (or middle, if you will), the angle of attack is very high -- often around 45 degrees. This angle of attack decreases as you go outwards, and at the tip it's much smaller.
If you didn't do this, and instead had a constant angle of attack throughout the entire prop, the `pitch' for a certain radius would be porportional to that certain radius. Yes, your plane could fly like this, but it would be far less efficient than a standard prop in normal flight.
The variable angle of attack is an attempt to make for a somewhat constant pitch across the entire prop. But most plane props aren't designed for creating static thrust -- dynamic thrust is the thing they're trying to optimize. If the plane is stopped, most of the prop is likely to be stalled, and so it'll be creating a lot more drag and a lot less thrust. (In case it's not obvious, stalled wings do still create lift, but they create a lot less of it, and a whole lot more drag while doing so.)
The lower the overall pitch of the prop, the less of the prop will be stalled if the plane isn't moving. So it may be that given standard airplane props, the pitch of the prop doesn't have a large effect on the static thrust due to this (but it probably does have a large effect on the power absorbed.)
Taking a prop where the angle of attack does not vary across the length of the blade, it's very obvious that the angle of attack (pitch, if you will, but on such a prop pitch is not anywhere remotely close to a constant) *does* affect your static thrust.
If there's any doubt, just whip out your helicopter (but make sure it's one that has collective control) and hit the throttle hold control and then adjust the collective and see what happens to your static thrust. In that case, the static thrust is probably roughly porportional to the angle of attack (assuming an uncambered airfoil. If it's cambered, then that'll be sort of like having a little bit of built-in angle of attack), at least for small angles.
In any event, I'm a bit surprised that the pitch ratings of standard airplane props have been found to have so little effect on the static thrust measured, but after thinking about it, it's not so unexpected. Our props are optimized for certain airspeed/rpm ratios, and outside of that they lose efficiency, and having zero airspeed (static thrust) is an extreme example of getting outside of what they're designed to be good at.
And as mentioned before, static thrust should only really matter to you if you're hovering. So helicopters -- they care. 3D planes -- they care. Everybody else, not so much.
Sure, you can come up with some extreme cases where you have a plane that has so little static thrust that it can't even fly, or a lot of static thrust and yet it still can't fly (because it has a super low pitch speed) but these cases are not the norm.