You don't say anything about the first sentence of my post? Can you point me out from where you have inferred that I don't know what I'm talking about, professor?
And your first sentence was
If the second derivative of the speed is positive, the acceleration of the cyclist always increases. That's not true
Power is a strictly convex function of speed, and is the integrand, to wit, the function to which Jensen's inequality is applied to deduce a conclusion relative to energy expended (integral of power) as a function of speed as a function of time. The result of which, under the stated assumptions, is that constant speed uniquely minimizes energy expended for a given average speed. In fact, even though the quantitative impact of departure from constant speed would differ, the same argument would hold via Jensen's inequality if the aerodynamic resistance were quadratic or quartic rather than cubic, such is the beauty of the approach. In fact, any exponent greater than one (whether an integer or not) would "do the trick" here.
Note that if power were a linear function of speed, then it would still be convex (but would not be strictly convex), and Jensen's inequality would still hold, but without strict inequality, and therefore the constant speed solution would still minimize energy, but would not be the unique solution to do so.