If we do need people to step it up, we have to ensure that we're not losing credibility. July has been a cool, wet month where I live, certainly not as warm as last year, so when we get articles saying "IT'S A CLIMATE EMERGENCY, HOTTEST MONTH EVER", and meanwhile I'm sending my kids off to school / daycare with sweatshirts in case the cloud cover sticks around... yeah. I can recognize the data, and I can believe that overall things are hotter, but it's not meshing up with personal experience, and so it makes it bloody difficult to, say, leverage my finances in order to buy an electric car (when my reasonably-efficient gas car is working fine and has decades of useful life left), and so forth.
"Climate emergency" rhetoric is simply going to burn everyone out and get laughed at, even if it's true.
I think we should focus more on changes that don't require individual action over and above the norm. Legislating a new normal is hard, but more productive than voluntary change.
To your point, it doesn't help that the weather - for local and national "news" - is used an attention and sensationalism tool.
The threat of couple inches of snow is The Rapture. The possibility of an above average rain storm is Armageddon. When these things don't live up to that hype the public subconsciously loses faith. Add in a complete lack of leadership (to get the public involved) and most ppl just consider it another "what can I do?" issue.
I don't blame them. I don't necessarily agree. But I understand why they feel that way.
This is tangential to your point, but given the carbon footprint of manufacturing a car, this would almost certainly be counter-productive.
Obviously, the incentive structure isn't set up to make that make sense for auto manufacturers, of course. Look at a Tesla, it's even less repairable than a normal car!