No, but we already do. Huge amounts of public money is spent by government on consulting firms, private contractors and industry grants/incentives.
Yet the public doesn't generally consider these firms to be publicly funded organizations, despite taxpayer money being the primary revenue source for many of them.
The government switches external firms as soon as another firm seems to be able to do it better, that isn't true for the governments own parts. That makes the two fundamentally different, one can accumulate bloat forever the other will get renewed from time to time. Private profit seeking adds overhead though so which one is better depends on the domain we are talking about, in some cases private are better in other government are better.
We certainly do have to, at least for certain values of “we”. An example in the US is the current legal obligation to procure private health insurance. (There are exceptions to this obligation, generally lack of means, and in turn may qualify one to procure private insurance with tax subsidies.)
I had to refresh my memory, so I looked it up. What I see now is that the mandate is technically still law, but the tax penalty was eliminated. Not that it’s a meaningful distinction, just an odd legal artifact.
Yet the public doesn't generally consider these firms to be publicly funded organizations, despite taxpayer money being the primary revenue source for many of them.