Without a warrant requirement, all patients collectively pay their privacy away without knowing it. Is the collective benefit worth it? (Remember, the "benefit" is the additional harm the government prevents when not constrained by a default warrant requirement, not the total harm the government prevents with and without getting warrants.)
Our entire system is built around the fundamentally American belief that adults need to obtain permission to consume anything that isn't Alcohol or Tobacco (Unless you're an Indian living on a reservation where alcohol has been banned as well).
American pharmacies take special care to put our full names along with our doctors names and phone numbers just to make sure that nobody gets away with possessing drugs that the aren't allowed to have.
A big country like Iran banning alcohol seems unimaginable until you remember that for whatever reason alcohol is the only thing a big country like America hasn't banned.