It's just more fun to yell about tax loopholes that corporations utilize, because big faceless corporations sound more evil than tax breaks that John and Jane benefit from.
But once you've established the deduction and people have already bought homes expecting it, they require its continued existence to be able to keep making the payments.
What they really ought to do is remove the deduction for new mortgages but not existing mortgages (or refinances of existing mortgages that don't increase the loan principal).
The result of this is that better off people underestimate how much direct government assistance they have received [1]. I won't comment on the potential political motivations for dividing up the welfare state in this way
[1] http://themonkeycage.org/blog/2011/02/08/the_invisible_ameri...
Also, even if it is not much of a win for the wealthy households, higher property prices generally mean higher rents so it disadvantages the class of people that rent. A similar argument applies to housing benefit in the uk, which is something you receive if you are poor (working or not). The recipient doesn't really see the money, but it helps to keep rents high so it's more of a 'landlords benefit', much like your mortgage deduction seems to be of more benefit to the banks than home owners.
The first problem you could address by capping the benefit somehow. The latter you cannot.
Or, if the government decides to tax everyone at 100%, but then gives out a series of "tax breaks" that lands everyone at their current tax level, is whatever government people get to keep welfare?
I don't want to go to far in the other direction. It's 100% possible for politicians to spend money via tax breaks. So neither worldview really works, but you have to have something between them.