what exactly is the advantage of going after their insurance vs. filing a claim with your own?
If you are made whole in the end it should make no difference to you. The legal principal is called Subrogation, your insurer can go after anyone they think caused them to pay you. Usually they don't bother because the cost of pursuing broke defendants isn't worth it.

The worst situation is when both drivers have the same insurance company. You are both then screwed.

If you aren't getting everything you think you should you ought to consult an attorney. I don't know anything about the laws in your jurisdiction, but in my experience the mere appearance of a lawyer will make a significant difference. This may help with the Sheriff as well.

I don't understand why the Sheriff is being obstructive, I'd think they'd want you to testify about the "drunk" "driver".

I AM NOT A LAWYER BUT DEAL WITH THEM ALL THE TIME.
MY ADVISE IS THEREFORE ILLEGAL.