Look at the booklet the IRS sent you. But as I read it, the limitation had nothing to do with the value of the home but with the gross income.
Prices rise to capture the subsidy.
Rather the prices rise until the demand curve crosses the supply curve. In the real world neither happens. As you agreed, the price rise caused by the MITD works exactly like the price rise in response to interest rate changes. Almost no one makes an offer based on the interest rate. If the price rise captured the subsidy, then no new person could enter the market because the rise in the price of houses would cancel out his additional ability to pay.