I believe this representation of the Euler-Lagrange equation might be confusing. I don't think that q and q' are independent just because we are taking partial derivatives. An alternative expression would be:
Let L(x, y, z) be such that L(q, q', t) = K - U. Then the Euler-Lagrange equation gives:
(d/dt) ((dL/dy) (q, q', t)) = (dL / dx) (q, q', t).
On the left: (1) Take the partial with respect to the second coordinate where L is a function of x, y, z. (2) Substitute in (q, q', t). (3) Differentiate with respect to t.
On the right: (1) Take the partial with respect to the first coordinate. (2) Substitute in (q, q', t).
I believe this is why q and q' are "unrelated" in the Euler-Lagrange Equation.