Just to be clear, there are two problems with the mortgage interest deduction. First, it's regressive; by the numbers, the people who benefit most from the deduction are those least in need of a benefit from the government. Second, regardless of your income level, the mortgage interest deduction artificially increases the desirability of residential real estate, which drives up the prices, which acts as a shadow tax on home ownership.
The first problem you could address by capping the benefit somehow. The latter you cannot.
My microecon is a little rusty but isn't the second problem essentially a wash? It seems that the shadow tax should never exceed the mortgage interest deduction (assuming rational markets).
To a first approximation, the MID pushes up the costs of housing by as much as the tax deduction. Making it entirely useless for "encouraging home ownership."
The first problem you could address by capping the benefit somehow. The latter you cannot.