Is the elevator really an 'angle of attack control' as described in Stick and Rudder?
Wolfgang Langewiesche is right to appropriate order of approximation. For every elevator position a statically stable airplane settles to a specific equilibrium angle of attack. That's how static stability works.
Airplane is statically stable if and only if increasing angle of attack causes higher increase in coefficient of lift on the aft airfoil (tail for normal layout). Then increasing angle of attack (by random fluctuation like turbulence) increases the lift aft more, the overall centre of pressure moves aft and the aircraft pitches nose down to return the angle of attack to the trimmed position.
By moving the elevator you adjust the coefficient of lift on the control surface, and the plane needs to assume a different angle of attack to shift the centre of pressure back to coincide with centre of gravity. Most aircraft are designed so this is quick enough that if you don't jerk the controls too much the angle of attack simply follows the elevator position.

isn't it true the Angle of Attack will increase because the relative wind is now coming from in front and above the plane?

The angle of attack is high when the relative wind is from below and low when it is from above.

If I then add power and climb (while holding the same attitude)

If you add power and leave the yoke as trimmed, the plane will start to accelerate, which increases the lift. That causes the plane to accelerate upward, which decreases the angle of attack. This decreases the lift more on the tail, so the tail sinks and the aircraft pitches up until it assumes the original angle of attack. At this point, the pitch angle is increased by the same amount as the flight path angle.
The aircraft is still moving faster than the original speed due to inertia, so the lift is still more than weight and the aircraft continues accelerating up and pitching up to maintain the angle of attack. When the speed returns to the trimmed, the aircraft is already pitched too much, so now it will slow down and pitch down again and repeat the cycle. This is called the Phugoid oscillation. In most planes it is damped, but still annoying if you don't counter it with a bit of elevator deflection. But if you don't touch the trim, just arrest the phugoid, the aircraft will settle in climb at the same speed.
It has some consequences:

Tendency of the plane to pitch down means it is losing airspeed.
If you are not pulling on the yoke, or trim aggressively nose up, you won't stall. Unfortunately pulling on the yoke to keep the nose up becomes too much of a muscle memory so it's easy to miss this clue.
It is how pitch for speed and power for descent rate works on descent.

See also How It Flies, specifically section 2.3 talks about trim and angle of attack, with more details further in chapter 6.