IT WORKS !!! FINALLY !!! AMAZING !!!
Version 2014.9.0.2021: Today, at 1:30pm, Central Daylight Time, I opened the avast Free interface and went to Settings > Registration. And, for the 300th time, I clicked the "Register" button (to the right of the words "You are not registered"). The next window that opened offered me the choice to "Select" either the paid version or the Free version. When I clicked "Select" at the bottom of the Free column, the thing actually took. My avast Free (at Settings > Registration) now says "You are registered." with 361 days remaining (to April 7, 2019; it appears that the 365 day re-registration period is calculated from the date of one's last virus definition update: my last virus def update was April 7, 2018; today is April 11).
NOTE: this was done when I still had 9 days remaining on the initial 30 day subscription following the March 20 re-install. I have no idea if the same would occur if I had let the 9 days run out and that initial 30 day post-install subscription period had "expired". Nor do I know if my version will, sometime PRIOR to the 361 days running out, again offer me a chance to re-register. I guess I'll figure those things out in about a year from now.