The key is that everyone feels qualified to judge these contestants and we are told that our opinion matters (they count our votes). The fact that Randy and Paula are punchlines has nothing to do with our belief that we are qualified to judge whether people can sing or not and whether they are entertaining or not and whether they are pretty or not-- the 3 things judged on AI. I don't watch dancing reality shows because I don't feel qualified to judge that. People also like to believe that a celebrity owes them for their success and AI taps into that. And AI makes these celebrities more humble because they have to literally beg us to stay on the show.
And people like to be entertained and they like to hear good singing and they like to look at pretty people, and usually that's what they get from AI. AI also tries to make sure that even bad singers are entertaining, to the extent that they can. Bad and boring is bad business.
AI also is smart to make AI seem larger than life so you feel weird for not following the show-- like you're a hermit. Big stage, big audiences, big ratings. bright lights. People are attracted to it like moths to a flame.
AI also keeps some information from us-- quite a bit, actually. They are playing hard to get and that tactic works in TV just like in romance. They don't tell us what songs the singers will sing, they don't give out vote totals and are even loathe to tell how people rank in vote totals. That makes people tune in to find out and it makes people gossip online about what they think is the truth about the things that aren't shared with them. People leak rumors and viewers eat it up-- just like political leaks and scandals.
People also question if the show is rigged because of this. If its rigged then our votes mean nothing and neither do the two hours spent voting in vain.