Well, couple things here:
1. No word yet on if he was "legally drunk". I understand some will say it doesn't matter, but laws allow for a certain level of alcohol consumption, based on studies of how it impacts impairment. If he's above the limit, then I agree with you. If he's legally under the limit, its hard to argue impairment is much of a factor.
2. The NFL really shouldn't have anything to do with this. They're not in the criminal justice business. This is a matter for law enforcement. If they take it seriously, then the NFL will have no choice to. If he's in jail or on significant legal constraints, then he won't be playing in games (if he were a player) or coaching in games (if he were a coach).
Like Ray Rice's punishment was exponentially worse from the Ravens and the NFL than it was from the legal system. We could argue all day long whether that's right or wrong or not, but the bottom line is... private, for-profit institutions are not qualified to be in the outside-of-business punishment business. They should be following the lead of the judicial systems in this Country who's job it is to adjudicate these matters. If they take it seriously, the NFL should. If they don't, then they probably shouldn't, unless it's directly impacting its business model.