It sounds to me as if the police handled this perfectly, and I have just a bit of experience in these matters.
Many do not realize that police officers are afforded zero flexibility in most of these cases. Attempting to run someone down with a vehicle is not something police officers are allowed to overlook, let alone much smaller offenses. Things changed across the country after the OJ case. In NYC, they changed drastically in a matter of a few weeks.
Now for some speculation: I would say that there is a 90% chance that alcohol was involved, which it is in a big majority of cases. He was arrested approx. 4 hours after the alleged incident, right? There was ample time for some of the alcohol to have worn off. Also, he could have decided to drink AFTER he left the scene, thus no arrest.
And, here is a little secret about alcohol:
If a person is a consistent drinker, it is almost a given that the most out of character, wildest things they have done in life were done while consuming alcohol.
If a person is a virtual non-drinker, the wildest, most out of character things they have done in life ALSO were done while consuming alcohol more often than not.
Now, they were all out until 1:30 AM. Jay, who happened to be at the home of his ex and sees her with a "friend" (which the "friend" thing could possibly be true). My guess is that it will be a bit dificult for Jay's lawyer to come up with a plausible defense for him, but of course I wasn't there.
The above is just an educated guess, but I will say again that it sounds to me like the officers on the scene and their supervisors did a fine job.
I also wish to compliment Jay for not fighting the police officers. He is probably a very tough guy and there would have been quite a few injuries if he had done so.