> $1700 per month pays off a $350k loan in 17 years, does it not?

No, it does not. You forgot the interest. Let's call it 6%, close to the current US average.

The interest by itself comes to more than $1700 a month!

Paying $1700 per month, you'll never pay off $350k, even with a 1000 year mortgage.

To pay in 17 years, you'd need to pay $2741 (plus fees) per month. Most of that will be interest at first, but it tapers down. If you want to start out not paying mostly interest, you'll need to pay at least $3500 (plus fees) per month.