We've used VBRO several times, but each owner can set their own rules - you deal directly with each owner, not VBRO itself. We've had great luck, good rentals and good experiences, but never had to put up with third degree.
While possible, I would consider it somewhat unlikely that a landlord would run a 50-state background check for each potential tenant. For one thing, it takes three weeks to get results back, and for another (we do this at work) they cost about $250 each.
While no harm in DL, the bottom line is that they will have your deposit which should be sufficient, along with a way to contact you.
You may have just encountered a landlord with a very low degree of trust, possibly been burned before. Can't blame them for trying to get every scrap of information, but you are just renting for a week or two.
As others have mentioned, I would move on to a more reasonable landlord . . .