That "you wouldn't download a car" became the meme while the ad itself said "you wouldn't steal a car" drives home exactly how effective these ads were at conflating the two.

Yeah, I guess so. Best I can recall, most people I knew were either avid pirates or casually saw it as low-level criminal activity by the early 2000s.

You've never seen this? Never heard "home taping is killing music" or "don't copy that floppy"?

I blame the "piracy is stealing" advertising/propaganda. It was super effective, given that we all remember it.

Editing to add one of my favorite videos in the other direction, Copying is Not Theft.