We got Serenity, the Peacekeeper Wars and Expanse S4, 5 and 6 out of fan campaigns, not to mention saving Star Trek TOS (different era though)
Myth. TOS wasn't saved by the letter campaign; it was on the bubble for a third-season renewal, and it got renewed for the same reason most struggling shows get renewed: because its producer agreed to reduce the budget for the new season, cutting the number of episodes and having fewer guest stars and fewer location shoots. The network announcement regarding the renewal and the fan letters wasn't saying "Okay, you convinced us to change our minds," but just "Look, we're renewing the show anyway, so please stop flooding our mail room."
Really, I don't think any letter campaign or petition has ever led to a show getting renewed in and of itself. The primary determining factor in whether a show gets renewed or a revival gets made or whatever is whether someone with decision-making power inside the studio or network
wants it to happen, and is able to make a convincing case for it. Strong fan response can help reinforce such a case, but it won't change the minds of powers-that-be who don't already want to do what the fans are asking.
For instance, the 1989 FOX TV series
Alien Nation had a strongly loyal (if not huge) fan following and critical acclaim, and almost everybody at the network and its affiliate stations supported the show and wanted it to continue. When FOX president Barry Diller decided to cancel it and redirect its budget toward making more half-hour comedies (because
The Simpsons had just become a breakout hit and he saw sitcoms as the future, and because you could make 3-4 sitcoms for the cost of
Alien Nation alone), others inside the network strove to find a way to continue the series, and a couple of TV movie scripts ended up getting commissioned. But Diller rejected the scripts without even reading them, because he just didn't care for the show. All the urgings of fans, critics, and fellow network decision-makers couldn't overcome Diller's opposition. Yet once Diller left a few years later, he was replaced by someone who was a big
Alien Nation fan, which is why the shelved TV movies finally got made and three more followed.