Updated.

Four televisions, three coming off an HDMI splitter, the fourth splitter output going to an AVR and then to the main family room television.

With the whole system on, I could turn off the television that is fed by the AVR, and after a few moments of black screen, the picture would reappear on the remaining three televisions. No further issues. That's what I wanted.

With the whole system on, I could turn off the AVR and same same. After a few moments, all is well.

It was when I turned off the AVR, and two seconds later turned off it's TV that the three other screens went black and stayed that way.

I changed my macro as follows:

HDMI2 to get the TV on the right input for next time
1 sec pause
AVR Off
8 sec pause (during which the FOUR televisions re-sync)
TV Off

The other three televisions don't go through a re-sync, and will remain on as hoped.