What about when I pay a lot of my mortgage early? Say, within a few years?
Paying straight cash for a house, or buying a house on a small fixed interest and a short term already baked into the mortgage, can gain you something. Paying off an existing mortgage early gains you time and the psychological benefits of not worrying about making the mortgage next month. If you're about to die, it lets your descendants take their time selling the house if they can't afford to make the payments.
Think of it in this way: as an investment, I'm making a secured loan to you from which I am expecting a given return over the next 30 years. If you're excited to pay me off in 5 years instead of 30, maybe we can make a deal (quite often there's refinancing involved in that arrangement) but I got into this deal expecting 30 years, not 5 years, of interest at that rate, and you agreed to that, too.
It's a simple calculation really, how many customers will actually pay off their mortgages early vs how many customers you will attract by offering the option to do so.
Any additional amount I send in on mine is credited as "additional principal payment" and reduces the amount owed (and hence the interest). If I had <mumble> thousand dollars in hand, I could pay my mortgage off Monday without any penalty, just five MONTHS into my refinance.
I have seen no-refinance-before-such-date mortgages, and while it's possible that a mortgage with a prepayment of additional principal penalty or barring exists, it's certainly not common in the US.
While a mortgage won't be structured this way, evidently an equity loan can, and this I do know from personal experience.
In the early years of a mortgage, the buyer is gaining the leverage of a large amount of the bank's money, so a huge percentage of his payment goes to interest on the loan. In the later years, when the remaining principal is smaller, he pays less interest. The buyer's total payment is the same amount over the term of the loan, but the portion allocated to principal vs interest varies linearly over time, til remaining principal == 0.
At any time, the buyer can pay off the note by delivering the currently-remaining principal to the lender.
The schedule you describe would effectively prevent anyone from ever moving before their mortgage was mostly-completely satisfied. If you bought a $200K house and sold it a day later, you would have to send the bank a check for about $600K. I'm pretty sure this never happens.
Furthermore, the lender is always making his margins (modulo interest rate variability, and ignoring loan quality), because any payments sent to the lender by the borrower will get re-loaned to someone else. I guess you could write a contract so that the borrower would be penalized if he paid off early, but I'm not sure it wouldn't be usury.
Note that if you double a mortgage payment, you should be explicit that you want the excess applied to your remaining principal. Otherwise, the bank can choose to apply it as "the next payment" and therefore split at your current ratio. Regardless, you still have to pay next month's bill too. :)
Its not uncommmon to pay ~2x the principal in interest. So I couldn't say for sure, but I don't see it having a huge impact.