Yes, some of what we're talking about is minutiae, and we're talking past each other. In my paragraph where I distinguished CL (current) from CLmax, I offered my impression of what the disconnect is. I'm still not sure what your thoughts on that are, and that would be helpful.
Here, I'm seeing another potential disconnect, which is that it seems you're objecting to the direction of causality - is it the speed change that drives the CL change, or the CL change that drives the speed change? Well, ultimately, it's not that important, and may work both ways in different real-life scenarios. But, kind of like the question of whether the Earth pulls on the Moon or the Moon pulls on the Earth, who cares? Richard Feynman's "shut up and calculate" gives the answer of how the objects will behave.
And the answer is that CL is absolutely, positively correlated with speed. From the most recent attachment, when that plane is flying at stall speed (100 knots), it's flying at 1.50 CL. At 110 knots, CL = 1.25, at 120 it's 1.05, etc.
If you command 12.5 deg. AOA with the elevator, which is the same as saying you're commanding a CL of 1.05, the steady state speed will be 120, period dot.
You gave some examples, but I'm afraid were not careful and mixed steady-state with dynamic conditions:
If you keep the same configuration, [and I'll add, same trim] and you speed up - you'll create more lift (as a function of V squared) and you'll climb.
You probably meant to say "throttle up." From that, you'll speed up
only temporarily and from that climb (lift exceeds weight, CL is unchanged) - until you reach the climb angle (after the phugoid has damped) where the new thrust is equaled by the drag combined with rearward component of weight, and lift equals weight (I think we can agree to ignore the cosine of climb angle tiny reduction) and is therefore unchanged - so Lift, CL, AOA, and speed are all at their original values.
Short version of the above: if you just throttle up, ultimately you won't speed up, you'll climb. If you want to speed up
permanently, that can not be achieved without reducing CL.