This is a very calculated strategy. His aides came up with this during the night when they realized his goose was cooked.
If they cry "prescription pain medication", they hope to squash any criticism of his illegal behavior and special treatment by the police because they will cry "Well, Rush was addicted too!"
He was drunk, but drugs may also have been involved. Two of the police present have said they smelled alcohol. A waitress at the bar IDed him drinking there.
They way he will get away with it is once in rehab, they can treat you for all addictions - booze plus drugs. Then they can justify the cover up.