Woooooof, there's a lot to dig into here. You and I could probably go all day.
I appreciate the thorough and robust response, and you're clearly a thoughtful person. But I don't agree with any of your conclusions here. With full respect to you, here's why.
From the top:
Newness of capital
Origin of capital is an issue separate unto itself, but the simple fact is that economic growth comes from new capital. Maybe that capital comes from new investors, or maybe it comes from the creation of new value, but new capital is the engine of economic growth.
New capital matters. Don't discount it.
New capital means liquidity entering the system, basically. Wealth accumulating.
For all of government's might and magic, this is one thing government simply cannot do. Government cannot create new wealth. All that government can do is reallocate confiscated wealth. It's a zero-sum game at best in a vacuum, to take wealth from one place and allocate it to another. In realize this reallocation is net negative, since the initial funds face immediate shrink via inevitable (and significant) ancillary bureaucratic expense.
growth potential
Growth potential is the only reason to have investment, man. Why would anyone invest otherwise?
But growth potential does not define an investment, only the motive for that investment. What defines an investment is staking accrued capital in the pursuit of future gains. In other words, investing means risking currently existing value in exchange for greater future returns.
That's where government funds and private funds differ. Private funds come from accrued savings. Government spending comes from taxes (economic theft) at best, and more often from just straight debt. There's no accrued value in either case, though one is more destructive than the other.
businesses that wouldn't exist without government
True that. So what? If a business needs government cash, it's not a wealth generator, either. It can look like one, and plenty of folks can get rich in an organization like that, but it's really just a derivative. It's a parasite that feeds off of a parasite. As long as there's enough capital flowing through the system, everybody looks like they're generating wealth, but if tax revenues wane, you quickly discover that the emperor has no clothes.
This is the reason that analysts so often accuse the government of inflating bubbles in the private sector. Quasi-private firms that really suck at the tit that sucks the taxpayer's tit can look like successful enterprises. But if you follow the thread back to the source, there's no net creation of wealth in many cases.
the government could run a business that creates profit
Probably. They almost certainly do now. But on the whole, the government is a money-losing entity, not a value-generating one. Isolated pockets of growth do not come close to balancing the books when set against the sunk assets buried into the majority of government programs.
You can get out if this one by pointing to the exponential private sector wealth created as am ancillary byproduct of some government programs. I'll agree with that. There have been exponential ROIs on a subset of government R&D, and that deserves to be celebrated. I don't diminish that. All I'm saying is those programs weren't investments, regardless of how they turned out.
Failed government programs weren't investments either.
multiplier effect & roads
Yup, see above. Never disputed that there have been great economic benefits from certain government programs. I like many government programs. But they're not INVESTMENTS, and that's been my only point all along.
crumbling privatized roads
I actually don't think privatized roads would crumble, but that has nothing at all to do with this discussion. I think there are some interesting models for how roads could be privatized, but I don't see much point for those. In our system, the government does a good job with roads, and I see no reason not to let the government keep on doing it.
We enjoy tremendous economic output from the roads and infrastructure. But the funds that for them are still not investments. They are reallocated resources. Still awesome, still valuable, but not investments. They are not funded against any stockpile of wealth at stake, and failure is not possible. No one will go broke funding our government roads, but no one will get rich either.
can an heiress invest?
This is just silliness. Of course an heiress can invest. She would be drawing against an accrued stockpile of economic value. Any bet made leveraged against current stockpiles of wealth is an investment, whether the investor is rich or poor. There need only be accrued wealth (meaning savings) at stake.
political outlook & I'm an idiot
I haven't been particularly political here. This is just textbook macroeconomics.
investment only means private investment
Excuse me for shouting, but IT DOES MEAN THAT. It means EXACTLY that. Where else is the capital to come from?
Governments do not create capital. (They can print money, and that's a discussion for another time.) Government money is reallocated private money. There IS no government money, anywhere, for anything, without private investment.