People don't think "car supremacy is being challenged," they think "my commute is annoying already and now it's about to get a lot more annoying."
I like bike infrastructure and use it frequently and I'm way closer to the yuppie-urbanist edge of the spectrum, but it's important to understand where people are coming from if you want to sell it to them democratically. "Car supremacy is being challenged" sounds to them like "fuck you and how you get around town," and it's unsurprising they react emotionally to that.