This is such a BS idea that needs to be killed with fire before it spreads. This is an idea floated by greedhound lawyers looking for money, any money, in lawsuits. If allowed, it would extend legal causality to such ludicrous extremes that nothing on the net would be safe.
First, in most jurisdictions, it’s illegal to be reading texts while driving. That should be the end of it right there.
Second, how does the sender know that the recipient is driving? Even if they did, the expectation would be that the recipient will either read it later, or have some safe hands-free method of dealing with it.
Third, why limit it to texts? My phone also will alert me if I get an email, a “Presidential alert”, when one of my Ingress portals is attacked, lots of stuff. Should someone unfriending someone in Facebook be legally responsible for a car accident?
Welcome to late-stage causality and old school lawsuit greed.