It wasn't that, as much as he had diabetes.
Although Rush has always said he quit before the tour because his diabetes made touring difficult. But in the movie, it's said he was fired, because management noticed he was getting tempted by drinking and other things, and they realized that drinking+road temptations+ diabetes was a bad combo.
I recall in an interview in the late 80's, Alex said he and John hung out all the time. But in interviews in the 90's, Alex said he hadn't spoken to John in 10 years.
Anyhow, his playing on the first album was pretty good. I enjoy the 1st album a lot, there are some really good songs on it.
But I suspect John wouldn't have lasted in the band even if he had done that first tour.
Geddy was already getting into more prog, and Rutsey was resisting, wanting to stay with the more straight ahead rock.
Geddy apparently wrote the 7/4 section that became song Anthem when Rustsey was still in the band, but John wasn't interested in trying to learn how to play it.
And then there is the story about Rutsey originally wrote all the lyrics for the the 1st album, but then tore them up because he didn't like them, and Geddy had to re-write all the words at the last minute.