I have read many forums which confirm its a very bad idea to put any money down on a lease. However, my wife and I decided to go ahead and do it. We had the cash a time of delivery and decided to throw it down to make out month to month easier. Knowing us, we would have blown that money on something stupid so we figured throw it down on the car. Like you said, when you do the math, they get that same amount of money anyway.
The argument I kept coming across was if you write off the car you loose that down payment so I guess its a risk.
Re losing money down if the car is totalled or stolen -
Depends on the insurance policy, specifically how the waiver of depreciation works. In my case I'm covered for the full MSRP / purchase price, regardless of whatever was done after that point re: capital cost reduction (what a down payment is on a lease).
That settled the matter for me and was the only reservation I had after running projections on the numbers. If the car is written off in the first two years I get back the purchase price of the vehicle, no reductions from depreciation or whatever was done on financing. Confirmed it a few times to be sure because I couldn't believe it as it went contrary to the typical comments on forums. Put another way, even if I paid off the whole vehicle immediately afterwards in cash, the insured amount is the original purchase price. You buy insurance on the vehicle and not on what payments you make. Put another way, when you total / get your car stolen and you're paying interest on your car payments, the insurance company doesn't compensate you for the interest you paid over the price of the vehicle.
Honestly doing the math and projections they get their money one way or another - either up front or over time in the payments. So my conclusion was it just boils down to whatever you prefer - put it another way would the car companies opt for some financing or leasing mechanism that would provide a massive discount relative to their other options? There isn't going to be some secret "free" plan.
I think people just say that on forums because they haven't actually run the numbers.