Why does $\frac{dq}{dt}$ not depend on $q$? Why does the calculus of variations work?

I find that calculus of variations could benefit pedagogically from a few more "dummy-variables".

Here's how I think about it: $L$, properly speaking, is a function on three parameters. That is, $L:\Bbb R^3 \to \Bbb R$ so that $L(u_1,u_2,u_3)$ is a number for any three inputs $u_1,u_2,u_3$.

We're interested in the function $L(t,x(t),x'(t))$. The Euler Lagrange equations should then be written as $$ \frac{d}{dt} \frac{\partial L}{\partial u_3}(t,x(t),x'(t)) = \frac{\partial L}{\partial u_2}(t,x(t),x'(t)) $$ and this is what the Euler-Lagrange equation is really talking about.

Certainly, if we wanted to compute $\frac{\partial L}{\partial x}(t,x(t),x'(t))$ with the usual definitions (or, I guess, with the "alternate interpretation"), we'd have some kind of chain rule to work through. That is, we'd have $$ \frac{\partial L}{\partial x}(t,x(t),x'(t)) = \frac{\partial L}{\partial u_2} \frac{\partial u_2}{\partial x} + \frac{\partial L}{\partial u_3} \frac{\partial u_3}{\partial x} = \frac{\partial L}{\partial u_2}(t,x,x') + \frac{\partial L}{\partial u_3}(t,x,x') \frac{\partial x'}{\partial x}(t) $$ However, this second interpretation is not the evaluation we're interested in.