Yes, that is the paradigm unfolding here. That is indeed one of the many critical attractions of self driving vehicles.
>The reckless drivers will shun self driving because they want to show off and have fun.
First, this is your unsourced head canon. People with drinking problems, cognitive impairment (both temporary and permanent), etc seem likely to be a far bigger percentage.
But it doesn't matter, because they won't get the choice, which again is one of the core aspects of self driving vehicles. Right now, human driving in America is a quasi-right because it's currently intrinsically tied to "personal arbitrary time/space mechanized transportation". I say "quasi-right" because while legally it's not formally a codified right or in the Constitution or something, as a practical matter if most of the population wasn't able to drive the country would fall apart. And that is recognized by government, both executive/legislative and judicial. Losing driving rights would be a major burden for tens of millions people and would disrupt their ability to get or retain jobs, in turn causing other societal expense. So traditionally government is quite hesitant to wield that stick and almost never preemptively.
Self-driving though would decouple human driving from someone having their own car and ability to have it take them where they want, when they want like right now. That in turn would completely change to political dynamics. Driving would truly become a privilege, and getting a license could become much more like class C licenses or professional emergency responder licenses or the like are. Much, much more serious training and testing, continuing ed requirements as with other professions, and much easier to lose. Because unlike right now, losing one's drivers license won't mean losing the ability to use a car. Anyone who acts recklessly at all, even if there is no damage of any kind, can just preemptively have their license rescinded until at a minimum remediation is completed. Because driving will just be a hobby.