Lorentz transformation derivation fails

Attempted derivations of the Lorentz transformation in the previous post here, which is similar to the light wavefronts approach here, do not work. The reason is that independent and dependent variables are treated alike, but they are not. I suspect this applies to all derivations of the Lorentz transformation.

Let us look at the first derivation more carefully. One does not need the light postulate of relativity theory for the following equations with the clock rate, c, which could equal the mean speed of light:

x = ct and t = x/c

for all rod-clocks. These equations convert a duration into a length and vice versa, but they do not convert an independent variable into a dependent variable or vice versa. If t is an independent variable, then so is x which equals ct.

If the first observer detects an event E at coordinates (x(), ), then a second observer who is moving at a velocity v relative to the first observer will detect the event E at coordinates (x′(t°′), t°′) such that

x′ = γ(x0vt°)

where is the independent time variable, x0 is the (dependent) x-coordinate if = 0, and γ is to be determined. Then with β = v/c:

x′ = γ(x0 vt°) = γ(x0 βct°) = γ(x0 βx°) ≠ γx0(1 − β).

The last step is an inequality, not an equality, because x0 since x0 is a dependent variable and is an independent variable.

The corresponding transformation for x in terms of x0′ = x0(t°′) is similar:

x = γ(x0+ vt°′) = γ(x0+ βct°′) = γ(x0+ βx°′) ≠ γx0(1 + β).

because x0x°′. The derivation fails.