All very valid points, however, paying off a 3% mortgage early is the equivalent of investing your money at 3%. If you can borrow the money at 3% and invest the difference at 10% (very possible over the long term) or even 8%, you might pay your house off even SOONER.
But it's all a matter of the borrower's/investor's comfort level with debt and/or investment risk.
I said this earlier, but I agree with acrozier22's logic assuming most people don't follow up and truly invest the extra money or see it that way.
I am in the school of thought that if you pay it off in 15 years, you not only save on future interest (and likely are more disciplined overall with your family budget), you also have zero house payments in 15 years and open up a new line of income in 15 years after being used to paying a large amount each month.
But, again, if someone is truly disciplined and does the math and says "I would have paid $900 a month, I'm going to instead pay $700 for a 30 year and now am investing the difference" the math works out....but I don't like debt.