No. The taxpayer pays $30 for a mortgage which has real estate as collateral. The loss is $30 if and only if the value of the property is $0!!
The lack of financial understanding in this country is really, really sad!!
Some of the assumptions you make might make sense were it not for fact that government is not in the business of making money.
I realize you are optimistic at the prospect of the government making a profit on this - I do not share your optimism is this regard and do not believe for one second the government will break even.
As for the government having real property as collateral goes - that is outside their purview.