In itself, it the tuner has I^2R losses to be sure, but if the antenna
radiates 100 watts with the lossy tuner in place and only 20 watts without
it, how wrong it is to claim an insertion gain of 80 watts for the tuner?

Very wrong!
Tuner is not attributing gain to nothing. It has losses due to losses in its
components, doesn't matter how you slice it.
When you match antenna to transmitter output (especially solid state 50 ohm
type) and make transmitter happy to produce full 100W output, you are not
adding "gain" from/by the tuner.
That transmitter when looking to mismatched load cuts power back to 20W, when
you insert tuner, you are not making up 80W in the tuner (gain?) but making
transmitter produce full output. You are now putting 100W in and getting some
90W out, 10W lost in the tuner, but antenna is getting 90W from the combo.
Where is the gain in the tuner?
Looks like things are getting ridiculous here. No tuner (passive) has a gain,
it has always loss.

Yuri