This doesn't even have to be an explicit thing. You don't need to threaten people to cut their funding, they will like the system the way it is set up, because they profit from it being set up that way. You can't be independent from something if you depend on it to pay your bills.
But then the rest of your message seems to contradict that statement.
If "healthy and diverse press" is only guaranteed by single source of funding (state) then I don't see how this is diverse or healthy or independent.
A democracy has a vested interest in keeping independent newspapers afloat. Subsidies are not shocking, provided that this is impartial ans according to well-defined criteria, and that it does not become the main source of income for the media.
France has the Canard Enchaîné, which is thriving and funded only by sales and subscriptions, although it is well integrated in the political landscape. There are other examples in other countries but far too few.
Because the choice isn't "government money" or "no money", it's state (democratic) money or corporate (undemocratic) money! Looking at things like the murdoch empire we can see which we prefer.