This seems like a really bad idea for all sorts of perverse incentive reasons. Primarily I'd guess related to the fact that the costs of running a city of a given size are basically entirely fixed costs, rather than being marginal. Especially when you add that most of the essential services a metro-area government provides are revenue negative. Trying to make (for example) the health department "pay fair market rents…
If the health department is not paying the actual price, someone else is paying the difference. The department therefore does not get cheaper, it just looks cheaper on paper since the real costs are hidden. The reduced transparency is what stands in the way of creating democratic or market based decisions on the departments funding.
Most public goods are public goods precisely because a majority of people will purchase them under the former conditions but not under the latter. There are a wide variety of public services which you personally would not be willing or able to purchase for the total cost of providing the good, but would happily go in on if the rest of a large population could be made to pay their fair share.
A number of die-hard libertarians believe this is theft, which it is in a way, but relatively few of them are willing to move to a remote location and live off the grid to avoid this. Perhaps this is because they know that the alternative is also theft (or worse), because the common defense and the legal system are also public goods, and so you have no way of enforcing your property rights without relying on taxpayer-funded systems.