Before our title win the last non big 6 side to finish in the top 4 spots was Everton in 2005. It was a miracle for a a non big six side to even get champions league football let alone a newly promoted side and for us to actually win the league was unthinkable.
In Forest's era, newly promoted sides (promoted in the last 2 seasons) would break into the top 4 every 3 seasons. The gap between the top teams in the first division and the top teams in the second division was a lot smaller than nowadays. I would actually argue that the gap between Man City and Arsenal/Spurs in recent seasons is bigger than an established 1970s team like Arsenal and a 1970s promoted second division side. It's certainly more of a common occurence in the latter for the weaker side to finish above the established side.