I still remember the days when (certain) yeshivos didn't allow bochurim to drive (more common in EY).
Basically every self respecting yeshiva in Israel doesn't allow the bochurim to get licences, without special exception for extenuating circumstances. It's also hard to get a licence under the table there without yeshiva finding out - the yeshivos have contacts in the Motor Vehicles and Licensing Department, who alert them when a licence is issued to an identity number on their watch list. (There's nothing shady or illegal about it as far as I know - the boys sign consent forms to this when enrolling.)
I don't know if they're anti driving licenses for safety reasons as much as for distraction reasons, though. Regardless, it's not realistic to implement such a model in America. Both American society and our transportation models and societal norms regarding vehicle ownership are very different.
Many American yeshivas do have rules about driving to or bringing cars to yeshiva.
