ETA: The only time you should every pay off anything quickly is if you plan on keeping it your whole life or it is an item that depreciates (like a car)
I disagree. Anytime you can pay a lesser price for something that's always a smart move.
$200,000 loan at 4% over 30 years will cost $343,000.
Put down as little as you can and spend as little of your own money as you can.
No, you're paying an extra $143,000 to borrow $200,000. Spending 171% of the purchase price to pay less today so you can to spend more tomorrow.
Now, take that same $200k over 15 years and you end up spending $266k, saving yourself $77k in the process, in addition to the equity that you've paid into the house.