The vast majority of "We Built This City is the worst song..." is just media regurgitation.
Americans love lists---"Best of", "Worst of"...doesn't matter. So when freelancers or interns (or Artificial Intelligence) get the assignment to produce a list of the worst songs ever, previous lists get scoured and "We Built This City" indexes high ---"Everybody hates it"---just because whoever it is that's compiling the new list sees it on all the other lists.
It's not that bad a song. Someone here educated me a couple of years ago to the fact that it was co-written by Elton John's lyricist, Bernie Taupin and Martin Page. They wrote it about a rash of live music club closings in Hollywood and L.A.
Not knowing that, I thought it made sense for Starship and San Francisco. There are people in the Bay Area who today will tell you that San Francisco was "built on rock and roll"---a narrow view heavily influenced by the late 60s explosion of the Jefferson Airplane, Sly & The Family Stone, The Grateful Dead, Santana and Creedence Clearwater, among others as well as some radio stations---KMPX, KSAN and KFRC---that achieved legendary status around the same time.
And---the line about "always changing corporation names", I took (again, mistakenly, because they didn't write it) as a gentle jab at themselves---from "Jefferson Airplane" to "Jefferson Starship" to just "Starship". Exactly the sort of self-own Grace Slick would have found delicious.
As for the DJ break---the voice is that of Les Garland, then VP of Programming for MTV, but prior to that (1977-1980), program director of KFRC.
Some stations played it without the DJ (especially AOR stations). Some played Les. Surprisingly, when KFRC added the record, they didn't play Les, but had then-PD Dave Sholin record exactly the same rap Les did on the record, but adding "610 KFRC" at the beginning and "on 610 KFRC" at the end...forcing him to deliver it much quicker than Garland did.
The thing that I think contributes to the bad reputation is the video---dear God, is it everything wrong with the 80s all in one package:
Being just a little too old to have been in the thick of the MTV demo in its first few years, I never saw the video until maybe 10 years ago. And over time, I've come to realize that a lot of songs from that era were bigger hits because of their videos than they might have been if they had to stand on the audio alone---and there are also pretty solid songs that people might remember more fondly if the video hadn't been so damn cheesy.