Since a portion of the social surplus generated by investments goes to consumers and not producers individuals in a free market are always under-incentivized to make investments and therefore a degree of investment must be provided by the state, correct?
No this is not generally correct especially because you are making strong general claims e.g. always and also draw a policy conclusion from is’s, which breaks a principle of Hume's guillotine.
If there are positive externalities, e.g. investment in non-patentable R&D/basic research, then companies won't be able to capture appropriate level of return on investment in such R&D. In this case more socially optimal level of investment can be potentially achieved by some government policy or non-market solution.
However, there are important caveats here;
Not all investments have externalities. Students often confuse positive/negative externality as any positive or negative effect on other people, but this is simply not true. Following, Mas-Colell Whinston Green (1995)
"Definition "11.B.1 An externality is present whenever the well-being of a consumer or the production possibilities of a firm are directly affected by the actions of another agent in the economy." .... "When we say 'directly,' we mean to exclude any effects that are mediated by prices."For example, building a chip factory might have very large positive effects on welfare of workers and even consumers. However, as long as these marginal benefits are reflected in the rate of return on this investment, there are no external effects and level of investment will be optimal, unless some other market failure is present.
It is not apriory clear that the investment has to be provided by the state itself. There are various institutional arrangements other than direct state provision. For example, patents are examples of such institutional arrangement that does not require provision of investment by state. Moreover, this is not necessarily relevant for R&D but can be relevant for other investments such as investments in common pool resources, Nobel Prize winning research of Ostrom shows that it is possible to have non-market but also private solutions for externalities or free rider problems etc.
These solutions usually rely on some community level social enforcement mechanisms, so they are not necessarily applicable to some global scale R&D investments, but they are relevant for investments on local scale.
You are trying to derive oughts from is’s which goes against basic rules of public policy. Public policy is not about making economy as efficient as possible. Goals of public policy depend on particular molar philosophy and what moral philosophy is “the correct one” to follow cannot be derived from positive facts such as what policy mix maximizes GDP per capita.
Depending on moral philosophy you might want to leave some externalities or other economic inefficiencies unaddressed. For example, in Sweden there is a “Pirate” party that has goal to reduce level of IP protection, and they pursue this policy because they believe this is moral thing to do. Yet such policy will lead to under investment in affected IP sectors.
Long story short, you can’t just a priory claim that optimal public policy is policy that maximizes total surplus or total utility etc.
Another problem is that the older I am the less my money will compound if I invest it reducing the incentive to invest, so I might blow my fortune on frivolities because the reward for investment will either be very small or arrive once I'm dead.
No this is not problem per se. People do not care just about return for themselves but also for their progeny. In overlapping generations models where people die you do not necessarily get sub-optimal level of investment.
Now to be clear this could lead to problems if you add into a mix some other behavioral issues or market failures. However, in itself aging of an individual and eventual death does not result in under investment.
Moreover, if the rich person spends the money on frivolous things, then that spending becomes other person’s income. This is macro 101. One person’s spending is another person’s income Y=C+I+G+NX. Then other people will either spend or invest that money.
In fact, during periods of financial crises when the propensity to invest drops, and investment may became for all practical purposes exogenous, such spending might be even beneficial for raising GDP.
You can see discussion of these Macro issues in any undergraduate macro textbook, for example see Blanchard et al Macroeconomics.
Another problem is that if I am rich money might become meaningless to me so I stop investing and trying to grow it. Prioritizing my own self-interest in a free market will massively lessen social welfare.
I do not think this is empirically very realistic. Even extremely rich people have typically very large marginal propensity to invest l. However, let’s use your behavioral assumption as thought experiment. This is not a problem per se. It could be problem in combination with some market failures but in well functioning market this is not a problem.
If they spend the money on consumption then it does not matter per se as discussed above.
If they refuse to spend the money at all then that will have to endogenous effect on prices and rate of return. It will be cheaper for other people to buy things, and more desirable for other people to invest.
There can be problems in combination with market imperfections such as price or wage rigidities, but then the root issue is the price rigidity, not “meaninglessness of money” to a rich individual.