I wouldn't count on that necessarily. Mass extinction and threat to all advanced civilization seems abstract, your own ill health seems more personal. https://en.wikipedia.org/wiki/Framing_(social_sciences) You often see news stories with a personal frame to drive them home, rather than a situational frame.
Though I don't know, indeed, you might have it right, it might make no difference.
Hundreds of millions ignore the obesity impact on their health. And that’s something you can feel right away. I seriously doubt people are going to pay attention to something so intangible.
I doubt they ignore it. People that are obese and fine with it are a minority. That doesn't mean it's easy to stop being obese. Compared to obesity, CO2 doesn't have genetic factors, the science on it is clear, there are easy solutions to monitor it. You could also do something like make people do a test in a CO2-heavy room and in a CO2-light room. On the other hand, you can't show obese people what it's like to not be obese, or non obese people what it's like to be. The two situations are very different.
Though I don't know, indeed, you might have it right, it might make no difference.