I mean, if they're looking at their phone a push notification that overlays the texting app should be pretty disruptive to the act of texting and immediately enforce it.

Actually, I'm not sure if the overlay is possible, but a phone call would be pretty intrusive and on speaker phone / hands-free would be ok.

The first reaction will be trying to remove that notification, which helps nothing.

It seems that the best course of action would be locking out the user from the ability to text while driving, because whatever signal the user is getting is going to add to the time spent not looking where he should, but when the user can not text while driving, this time will always be zero.

(at least in regards to texting)

Yeah, but we're talking about the voluntary system shown. If they are just removing the notification, then would they have installed it in the first place?