I think we're arguing minutia here, but it's misleading to say that CL changes with speed - it doesn't. If you keep the same configuration, and you speed up - you'll create more lift (as a function of V squared) and you'll climb. If you're talking 'straight and level', it's not the CL that changes with speed change - it's the flight profile that changes (specifically the AOA), which in turn moves the CL to a different point on the AOA curve.
You can see how at the faster speeds, every degree of AOA yields a larger speed difference than at the lower speeds.
Yes, lift is a function of speed squared - but not because the CL changes. It's because the dynamic pressure is a function of speed squared, and lift is linear with dynamic pressure.