Well, a problem with the Austrian framework in this context is that there are things we need to finance as social goods (desired outcomes by society at large) that can never be structured to be profitable.
This is, incidentally, the classical economic case for both governmental provision of public goods, taxation, and regulation.
You can take the idea of non-coercion and private property pretty far, without even considering the role of charity. See Rothbard's Ethics of Liberty(1) for ideas of how it might work.
Also people are naturally motivated by more than commercial self-interest. This is why charities exist.
This is, incidentally, the classical economic case for both governmental provision of public goods, taxation, and regulation.