Let's say that you have two explicit (one dimensional) quadratic Bezier curves:
$f(t) = A(1-t)^2+B(1-t)t+Ct^2$
$g(t) = D(1-t)^2+E(1-t)t+Ft^2$
Where $A, B, C, D, E, F$ are scalar constants.
Then, let's say that you evaluate $f$ and $g$ for some value $t$ and linearly interpolate between those two values by some amount $u$.
Is the result you get always going to be the same as if you evaluated the function below?
$h(t) = G(1-t)^2+H(1-t)t+It^2$
Where
$G = A(1-u) + D(u)$
$H = B(1-u) + E(u)$
$I = C(1-u) + F(u)$
In other words... is lerping between the same time $t$ point on two curves always the same as if you lerped the control points, then evaluated the resulting curve?
It will be. To see this, just compute the linear interpolation explicitly, and then gather up all the terms that correspond to each of $(1-t)^2$, $t(1-t)$, and $t^2$. You'll find that they're precisely the $G$, $H$, and $I$ you've defined.