Honestly I didn't even recognize what song it was in the live version, or the band name, or her name for that matter.
With the band not even really being a one-hit-wonder in the US, despite being deeply invested in music and as an '80's teen, until I watched the video of the studio vesion, I had ZERO recognition.
I wouldn't call this "easy listening new-wave light" stuff '80's cool. Not when we had the likes of Bad Religion, The Cramps, Dead Kennedy's and so forth in the mix. Not unless you were pushing mid '40's in the '80's or were a lotal L7. This song is softer than the stuff my old man listened to.
But that could just be an attitude difference stemming from regionalism, and favored styles of music... much less one's definition of what "cool" is.