City didn't play United much (if at all) in Ferguson's first 2-3 years at the club, because we were in the 2nd division. The first (or at least the first meaningful) derby under Ferguson's reign was the 5-1, a game he described in his autobiography as the most embarrassing & humiliating defeat of his career. (I think there was a worse one towards its end.) When they beat us 5-0 a few years later he sympathised with Brian Horton, noting - from personal experience - that he would be goaded for months. He recalled barely being able to fill up with petrol without some blue shouting 'hey Fergie, what's the time? Five past you lot.'
He did not take that humiliation lying down. First, in the 90s/00s when they used to beat City, which was frequently, he used to trot out the line that it was basically three points, it wouldn't define their season (like in those days a derby win defined ours), they had bigger fish to fry.
And he added that United's real derby was with Liverpool. Again, in his 'everything I say is for a reason' way, he had stated at an early stage of his tenure at the swamp that he wanted to knock them off their fucking perch, which he did. In spades. But like you say, before the 90s really Liverpool v United was just another match. Sure, scousers v mancs was a big thing, but that was just as true of us v dippers (when in the top division) or everton as it was rags v Everton.
That 5-1 defeat had a lot to answer for, over the years, or to put the same point slightly differently, it explains a lot once you understand the psychology and the personal history of a bully. It wasn't enough for them to give us a pasting, it wasn't enough to humiliate us the way he had been humiliated. His revenge was to marginalise us. That fucking banner didn't go up by accident.
This is not to say the Liverpool v United 'rivalry' has largely wholly manufactured by Ferguson or even by Sky, who have been perfectly happy to adopt Ferguson's narrative because it suits their purposes to have millions of plastics across Ireland, Scandinavia and the far east to be invested in a game that is, today, 3rd v 6th (which is not only what the table says, when you look at how Liverpool were against us and Chelsea, is about right in terms of quality, too. Maybe you'd have the rags in 7th or 8th given who they've played so far.) This was, and always has been, a big game, and for all that Sky blow smoke up this game's arse, the vitriol that will be on display is genuine.
But, as you say, there is a reason why your average match-going rag sings silly songs about us and not about them. There was a time under Keegan 20 years ago when our major rivals for the league were Wolves. We still sang songs about David Beckham. Singing their nursery rhymes is the way they keep themselves relevant, even though in reality they no longer are.