The people you are worried about are not the ones who are worried about making the mortgage. The people you’re worried about are the ones for whom strategic default is looking better every day. In some areas, property values are still sliding. Homes are going deeper and deeper underwater. For some people who can easily afford their mortgages, financially, it makes more sense to get foreclosed on and take the credit hit. The deduction isn’t really all that big of a deal, but it is one more thing keeping those people from saying “screw it, I’ll rent.”
But there’s no evidence that it does increase home ownership rates. Mortgage interest isn’t deductible in either Canada or the UK, but the rates of home ownership are about the same as in the US, hovering around 68% as set out in this wiki article.