You're right about non-musicians not being bothered what instrument is doing what. It’s particularly strange the way musicians are obsessed with what other musicians are doing, especially if it’s not in the traditional way.
I think it depends on how prominent a particular instrument is. I was reading recently how in experiments non-musicians and singers were asked to sing a song. They were incredibly accurate with regard to both the tempo, and the key and pitch of the notes. Even though they couldn’t actually sing the pitches, they knew mentally where the pitch was. They were remarkably accurate with the tempo.
I think most people who know Michael Jackson’s Billy Jean for instance, would be able to sing; however badly, the opening riff. The same probably with Lou Reed’s walk on the wild side. A non-musician probably wouldn’t know what’s going on in the bass line, but they’d probably have a good go at singing the ascending/descending tenth figure.
It’s possibly the same with My Girl by the Temptations, Walking on the moon by the Police maybe too?