Ash Vs Evil Dead was produced by Rob Tapert, Sam Raimi, and Bruce Campbell. (Rob is really the show-runner of the three. Sam sticks to Hollywood movies, while Campbell is mainly the actor). Rob had a good relationship with Starz since they did the Spartacus series together, which was pretty successful. So Starz was a perfect place for Ash Vs Evil Dead, which benefited from being unrated on Starz. They could do whatever they wanted with it (little to no censorship).
So they make the first season and screen the first episode to a Comic Con crowd. The buzz is huge. Starz greenlights the second season before the first airs. (I think they were going to green-light it anyway though because they have a schedule to uphold and that's just how they work, I guess. Greenlight the first two seasons and then feel it out from there.) They green-light it around August or September (they always do things behind the scenes way in advance of public announcement - like pickups & cancellations, for example).
So they had this schedule of writing the season around December, filming around April/May, and then broadcasting around October-December. That's how all three seasons were done. And that's how a fourth season would have been done as well, obviously, had they continued.
Season Two was greenlit before Season One premiered in October (31st).
Season Three was greenlit before Season Two premiered around October.
But right around the time season three was greenlit, Starz was bought out by Lionsgate. Those Starz people in charge, that Rob Tapert had a good business-relationship with, are suddenly no longer going to be in charge. I think if they had stayed in charge, they would have made two more seasons to finish out a nice five-season series.
The ratings - Season One had alright live ratings. Season Two's ratings declined from Season One. But they didn't see Season Two's ratings overall until it was the end of December, the same time they're in the middle of writing the next season. (So that info from 'enorl76', about it being known it was over in the middle of writing the third season, is likely accurate).
Lionsgate was taking over at this point. Lionsgate saw the ratings and clearly made the call that it was over. They weren't buddies with Rob Tapert and were strictly about ratings.
Despite the show's popularity on Bluray, which had wonderful sale-numbers, the live-ratings (which somehow matter most) simply weren't sustainable for the series.
The third season's broadcast airing was delayed by about 5 months later than regularly scheduled (rather than an October 2017 premiere, it got a February 2018 premiere).
Being that the crew would be on an annual schedule and they would need to be notified of a next-season-pickup, around the same time each year, so they can be assured work or be released from their contracts to find other work, it became painfully obvious (to anyone that knows how this works) that it was secretly (non-publicly) canceled when December had arrived and there was no public announcement of whether it was greenlit or not.
The reason they couldn't publicly announce that it would be the last season is because they wanted as many subscribers as they could get for the third season's airing. If fans knew it was over, they'd be pissed and unsubscribe from Starz before the third season aired. So actors promoting the last season have to lie about it. Though, they did clearly hint that it could be over.
When Rob Tapert knew it was over, he brought up the possibility of a movie-continuation to Bruce Campbell, but Bruce decided that Season Three's ending was a fitting conclusion for Ash and that he no longer wanted to play Ash, ever again. I don't know if there's some unspoken beef between him and Rob or Sam, but from what I can deduce (from what Bruce has been saying) is that he simply doesn't want to be stuck in a contract anymore of having to play and promote Ash for years to come. Just one movie would mean (at least) another two years of prepping, shooting, and promoting. And in a contract, if that one movie was successful, then there'd be some clause for having to do a sequel or two. That could be a possible addition of four years on top of the two. It's a physically demanding role. And he's over 60 years old now. He has a lot of personal projects he's wanted to do (as well as the simple luxury of free-time) and wouldn't have that time if he kept devoting his life to Ash, of which he's already given so much.
Rob Tapert has a little hope that Bruce might return to Ash someday, just because he knows time changes one's mind and as time passes, Bruce may become bored with whatever he's doing and decide to grab ahold of the boomstick once again.
So they probably would have made some direct-to-video movie(s) if Bruce Campbell wanted to. But he had become a lot like Ash in Army Of Darkness: "I'm through being their garbage boy. I did my part, now I want back, like in the deal." "I got a chance to go back home and I'm taking it."
So, when they were shooting the third season, they knew it was very very likely over, that this was it. Though, they still had some months after shooting it for something unexpected to happen, until the annual time for renewal came around August/September. Lionsgate was (and still is) trying to sell itself to some bigger entity. So, for all they knew, there was a very small chance of some company, like say Netflix, coming in and buying Lionsgate for their gigantic library of films, and having interest in continuing Ash Vs Evil Dead.
My point is that they weren't totally lying when they said that they knew, while filming, that this was likely it. They simply knew it was over if some other deal couldn't be made.
Starz did sell the airing rights of the first two seasons to Netflix during the delay of airing Season Three, simply to try and entice a bigger audience to get hooked and subscribe to Starz for Season Three. So Netflix audiences that got hooked on those first two seasons either had to get Starz or wait for the Bluray because Netflix was never going to get Season Three from Starz.
Ash Vs Evil Dead was very successful on Netflix and Bluray, but for Starz (Lionsgate), live ratings are what really mattered. They could have allowed Netflix to continue the show, but Starz didn't want Netflix to profit from their show. In other words, Starz would rather kill Ash Vs Evil Dead than let Netflix make sweet sweet love to it.
As for Season Three's Starz viewership ratings, it was very bad. Mostly because during the delay in airing Season Three, Starz was dropped from a major cable-provider, which meant that somewhere from a third to half of it's national audience lost access to Starz.