Constant speed propeller angle of attack and thrust?

I find the aerodynamics of fixed pitch props relatively easy to understand -
the angle of attack varies with both forward speed and RPM.

However, something puzzles me about constant speed prop thrust. I realise
that the blades twist to give a different pitch, and therefore when you
increase MP to maintain the same RPM the blades will be at a higher helix
angle. However, won't the angle of attack (calculated by the relative wind)
by the same? Isn't that the idea? If the prop is at the same angle of
attack, why is the thrust higher? Once again, I know the pitch is higher,
but isn't the angle of attack the important thing in generating lift (which
is what thrust is) rather than simply the pitch angle? I've seen texts that
say you get a bigger bite, but that doesn't seem to explain why that matters
since lift is predicated on angle of attack.

Anyone have a good explanation?

TIA