It doesn’t work that way. Advertising is not 100% transactional, and “did this convince you to buy from us right now?” is not the only reason why companies advertise.
That jingle helps keep Farmers top-of-mind for consumers, and does very likely help build a positive image of the company/brand in people’s minds (though, clearly, not in your mind). Thus, when a consumer decides, “hey, I’m not happy with my current insurance company, maybe I should switch,” Farmers is more likely to be one that they might consider.
I can guarantee you that they have done extensive testing on that jingle and their advertising campaigns (both qualitative research (e.g., focus groups), and quantitative research (e.g., surveys)), and that, overall, they are meeting their marketing goals for the jingle and ads. You find that jingle to be obnoxious; I can also pretty much guarantee you that, among consumers who have auto and home insurance, you are in the minority in that opinion.
If the jingle wasn’t working, and if most of their target audience hated it, they would not have stuck with it for over a decade now.