It's just a balancing act. More down = lower monthly. Since I'm mostly retired these days I prefer to give 'em chunk up front when I have the money sitting there and keep the monthly lower as my income will vary from month to month. I could have done zero down but the monthly would have gone up appreciably. That's why I use the sheet, I can tell them to give me a price with different (or no) amounts for a downstroke and look at all the options at once. And again, in the end all that really matters is what the car ends up running you total for the duration of the lease since you don't care about depreciation or the long term value of the car, outside of how that factors into the deal you negotiate from the dealer's perspective.
And I suppose you can look at the interest you could make on the money you don't put down up front. In the stock market right now 4 grand over 3 years might be worth it, though that is of course a gamble. But just keeping it in a savings account you'll probably lose money after inflation as interest rates are crazy low. My savings account is paying .02%. Even a with a 3 year CD you'll be lucky to get .5% at the moment.
It's an interesting trade off right now. With mortgage rates at the unbelievably low percentages they're at the banks don't pay jack on savings accounts and CD's , which make sense. But I'd certainly rather pay 4% on $250000 over 30 years instead of having a $20000 saving account paying me 3% a year. My first mortgage was at about 10%, and I remember my mom telling me she paid 19% back in the 70's on her first condo. But then again, I'm pretty sure that condo cost less than my Mazda.