I've been struggling with some fridge logic in the wake of the series finale and need to discuss it somewhere, so spoilers for the whole series follow:
Never meeting Cassie would mean that Athan was never conceived, but that wouldn't actually unmake the Army of the Twelve Monkeys because Athan was never The Witness to begin with; the Monkeys' cycle would just end up absent their false witness, but Olivia's cycle would still exist in some form and so so would the Monkeys. Not killing Leland would have no real effect on time, because Leland wasn't to blame to begin with, and the virus would still end up released by the Monkeys directly at the incident in the Night Room, or in Chechnya, or by Jennifer, or one way or another, if for no other reason than that the Monkeys would manipulate time to ensure it (for some reason, see below). They could time travel to try to counteract all of those threats, but them being threats in the first place would not directly be Cole's fault.
Not trying to stop Leland from getting the source of the virus in Tokyo in 1987 would change things, in that Leland would have declined the sale like he was originally going to until Cole's intervention, so the virus would not end up having been refined from the source by Leland's company. But the source would still exist and still be up for sale and end up in someone else's hands and eventually get released one way or another, if for no other reason than the Monkeys would manipulate time to ensure it (again, for some reason, see below).
The source would have to never have existed in the first place to prevent the virus outbreak entirely. We the audience know that the source is future-Olivia's body send back in time accidentally during her defeat by the protagonists, which only ended up happening quite the way it did because of Cole's travels. So the source existing in the first place is kind of Cole's fault, in a really roundabout way. If he didn't travel at all, those events leading to Olivia's defeat never would have happened, and there would have been no outbreak; but instead, Olivia would have succeeded in destroying time, which is the real threat that Cole is supposedly responsible for anyway, if he is "the demon" from the fable.
For Olivia to be Cole's fault, her creation would have to be because of his travels. She does exist very indirectly because of him, I guess: he was the first successful test of Jones' splinter machine, which was used to send the Messengers back in time, one of whom became the mother (well, egg-donor) of Olivia. But if Cole hadn't been the successful test case, sure someone else would have been, and it's the existence of the splinter machine that's responsible, so Jones is really more "the demon". But let's say that for some reason without Cole as a successful test case Jones never got the machine working. You might think that then the Messengers couldn't travel back, and Olivia's conception (well, creation) would never have happened.
But why would the Monkeys even need Jones' splinter machine at all? They have Titan, and splinter-suits. If Olivia just wanted to ensure her own creation, she could send her mother-daughter back for explicitly that purpose by other means. Which then raises the question of why the Monkeys need the plague to happen at all. Supposedly the purpose of the plague was to spur Jones to finish her splinter machine so that there would exist a means of time travel with which Olivia and the Monkeys created themselves, but they were already funding the development of time travel before the plague happened, and built Titan with the help of Elliot in the past in circumstances completely unrelated to the plague.
It seems like there is no reason why, absent Cole's time travel or Jones' machine, Olivia wouldn't just arrange for her own creation by sending her mother-daughter back in time with a time machine she just paid Elliot to build (Titan), and then used her 'normal' daughter Emma to spy on Elliot to turn Titan into the time-ending paradox machine, no plague required, no Cole required.
So it's not clear why, in the timeline where Cole got erased from history, that's not what ended up happening.