If it was so terrible, why did it get played so often?
When I think "terrible", I think of the movie "The Room", which has been called the Citizen Kane of bad movies. It is truly awful. The dialog is awful. The acting is awful. There are all sorts of logical inconsistencies where things inextricably appear and disappear. By any metric it's bad. It's so bad that it has a cult following because it's so bad.
"We Built This City" just isn't anything like that.
Compare it to My Humps [1] or What Does the Fox Say [2] or Friday, just to name a few? These are a few random songs that are all miles worse than We Built This City.
So your argument is basically just a no true Scotsman.
I grew up listening to it. I still enjoy it. I also still enjoy Rock Me Amadeus, and Kashagoogoo's Never Ending Story theme, too.
I was ten. How old were you? I ask because your recollection has strong "seething teen" energy.