"However, unless you have an all-moving wing, how do you change your AoA without adjusting your climb angle?"
You are confusing deck angle with climb angle. The climb angle will be 90 degrees, but the pitch angle will be less.
I now see that Rainboe with some additional elegance pointed out the exact same thing in the next post.
"If you adjust your climb angle to other than 90 degrees, you then have the probem of a thrust component not in the vertical direction, which has to be balanced by a force perpendicular to your trajectory."
Yes, and this compensation will be some residual lift. However, in this discussion we have been careful to specify that we assume the thrust to remain parallell to the direction of flight, a reasonable simplification to make in a discussion of basic principles of flight.
"All I'm trying to say here is that for most applications, this straight vertical climb we're talking about does produce some wing lift."
Not unless you factor in other forces in the aircraft xz plane perpendicular to the direction of travel it won't.
"Nobody is saying the wing does not produce lift in a vertical climb."
Yes, I am. As long as we stick with the basic assumption above.