Does the bank automatically have insurance coverage on a new car because the dealer let's you drive the car home before getting your own car insurance

Not unless the loan document you signed said the bank would do that, and it's my guess there's nothing like that in the papers.

I don't ever recommend driving without insurance, other than in a really dire emergency.  "I was going to get the insurance tomorrow" doesn't count for anything at all, in court.  All it takes is a little bad luck, or a moment's inattention, to turn what seems like a small risk into a very expensive mistake.