>"the state is a more efficient investor of capital."

Do you have evidence for this claim?

If this is the case, shouldn't the countries with the biggest governments (as a percentage of total economy) also be the ones which grow the fastest, and if not, why? How can states be so efficient when they never admit to making mistakes, (almost never) give up on failed projects, or go out of business like private investors do?

There's no evidence needed because it's a political opinion. Efficiency doesn't boils down to having the biggest GDP growth, sometimes it's not at all about the money. The HDI is probably a better metric.

Do you have evidence that states with higher government spending as a percentage of the economy have a more quickly increasing (or even a higher) HDI? It seems that efficiency of spending would influence the trajectory of your metric of interest more directly than the metric's current level (this seems obvious to me, but I can explain why in detail, if required). I can think of a few examples which would support your case, and many counterexamples, but would be happy to hear your case.

I would also think that a system's efficiency would affect the economy as well as all other indices of interest (though perhaps in varying proportion), as a healthier, happier citizenry or one with more capital goods, and better investments would grow economically as well as in other metrics.