What's the complex Poynting vector in the tuner? Why should I believe
that the delay through the tuner is negligible? Why should I believe
that the stored fields are negligible? It's not obvious to me.

For that matter, why does the ideal autotuner not reciprocal? The
impedance in the presence of the standing waves at the tuner-line
junction is R+jX. That is transformed by the autotuner so the source
sees 50 ohms, but that also means that the output impedance of the
generator is transformed as viewed by a wave coming down the line.

It's not transformed to infinity. If it is a circulator and load, this
is a distinctly nonreciprocal system. If it's a tuner affecting a
match, it is reciprocal. If your definition of an ideal autotuner
includes the nonreciprocity, please send me a schematic of such a
device.

I think I calculated the Poynting vector for this situation, I think
the Poynting vector divided by the group velocity of the waves and
integrated over the area of the coax gives the energy density per unit
length in the steady state, and if you've tuned the line length so that
the imaginary part of the Poynting vector goes to zero then the
explanation you give falls apart.

The tuner matters here. This is the problem with English language
discussions, especially disagreements, about electrical systems.
"There exists an ideal autotuner at the input" is insufficient
specification of the mathematical problem. If you place a device that
does everything you say it does at the source-line junction, maybe you
do get 300J in the one-second line.

Why isn't it reciprocal? Why is its phase shift negligible? Is it
just because it's physically small? I seem to remember a certain hotly
contested paper that I would think would have certainly put that sort
of idea out of your head. I think the crux of the misunderstanding is
in the tuner. I think it has stored energy and nonnegligible phase
delay, else it is non-reciprocal and dissipative (circulator+load).

What *IS* an ideal autotuner? I think you've stuffed too many
conditions into its operation for it to be a physically realizable
device. I'm pretty sure that it breaks the time reversal symmetry of
the system. Send out a pulse into your one second misterminated line
via your ideal autotuner. Wait half a second. FREEZE! Now run the
film backward while obeying the rules of your autotuner. The pulse
doesn't go back into the source. It bounces. What's up?

73,
Dan