No, sadly they don't. Yes shows are sometimes picked up by another network or go to streaming, but it's not because of fan protests.I loved The Family! It had a lot of twists and was really well done. Super bummed about that.
And I'm obsessed with Nashville so I really don't get that one - people are trying to protest but does that ever work?!
There's a lot that the average viewer doesn't understand about ratings and the decision to cancel shows. If anyone is interested in understanding more of it, I recommend The TV Grim Reaper (formally known as the Cancellation Bear). Throughout the season, he predicts renewals and cancellations and he explains why. He has a really good record. The best at predicting that I know.
For example, this year's results for him were:
CBS: 21-2-1
ABC: 22-3
NBC: 18-0-2
Fox: 14-2-2
CW: 8-2-2