Gamma=1/sqrt(1-vv/cc) (he called it beta, but tradition now
calls it gamma) appeared magically in Einstein's t' and
x' transforms, replacing the mundane 1/(1-vv/cc) without
cause, reason, or justification.

But Einstein did cause it to appear in expressions for
the transformed y and z axes.  All he had to do was say
light movement along these transverse axes was at the
rate sqrt(cc-vv).

Remember, the (c-v) and (c+v) expressions Einstein used
were not due to non-c light velocity, but due to the
movement of objects toward which the light was moving.
That condition does not hold in the y and z directions
in his derivation.

"In an analogous manner we find, by considering rays
moving along the other two axes, that

    Y' = c*tau = ac(t-vx'/(cc-vv))

when t=y/sqrt(cc-vv), x'=0."

When x'=0, we find  that Y' = c*tau = act, just as every
SRian in the universe agrees.

In any case, the t=y/sqrt(cc-vv) line is the full,
ridiculous justification Einstein gives for the
existence of the expression sqrt(1-vv/cc).

Ridiculous?  Sure, x'=0 is a rather small subset of
the possibilities for x'; how do you generalize to
the full range of the universe from x'=0?

And there is not even the hint of a justification for
replacing (1-vv/cc) with its square root in his time
and space (x) transforms.

QED: Einstein's SR time transform derivation is invalid
by reduction to the absurd: it is eithered based on the
premise that x'=0 and not x'<>0, or based on nothing.