You misunderstand the fundamental mechanism for improving standard of living: productivity gains. Consumer spending does not improve productivity. Rather, investment in capital goods (by those rich hoarders) leads to productivity growth.
That's a fairly basic liberal view, profoundly flawed. Consumer spending improves productivity as a second order effect. Consumer spending improves the demand side of the economic cycle, prompting opportunities for investing in efficiently supplying that demand.
The disruption of the demand side of the economic cycle is the reason behind the loss of competitiveness of economies with high inequality. You get capital accumulation that finds no outlet for investment, other than riding economic bubbles.
If you're considering "investment" as included in GDP, it's not just "saving money", it's "business spending on depreciable assets", essentially. Owning a share of Apple doesn't make you "an investor" in the GDP sense, but Apple does a lot of investment.
I'm not sure that "rich hoarders" have an appreciable positive impact on business investment in depreciable assets. Most capital assets are purchased to produce goods and services to meet demand, which might be stimulated more by redistribution.
But those productivity gains need to be directed efficiently, which requires some mechanism for the market to know the needs of a broad swath of the population. People get to vote on what problems need solving and that directs productivity growth as capitalists chase revenue.
However, if income is too highly concentrated, then folks with serious problems don't get many votes on what problems need solving and the folks who do get those votes are disconnected from those with more dire problems.
Owning stock is not investing in economic terms. Companies are going to maximize profits independent on how much individuals pay in capital gains because nothing about capital gains changes the companies costs or profits.
Buying existing properties is the same thing. If no assets are created then it's not an economically useful investment.
VC money is often thought of as rich people investments, and it is an investment in economic terms. However, Pension funds, endowments, charitable foundations, insurance companies, sovereign wealth funds, and large corporations are more often than not the source of these funds.
Generally rich people are rich because they have invested their money they don't simply sit on piles of un-allocated resources.
Buying stock is investing in economic terms if it's a new issue. Buying bonds is investing, too (again if it's a new issue), and the bond market is twice as big as the stock market.
A relatively small amount of new stock is issued each year on the open market when you exclude companies that trade their own stock. Bonds are often issued for things like taking a company private which are again not really investments just asset transfer. Further, profitable companies often issue bonds even if they have cash on hand or regular dividends.
So again it might be a useful investment, but that’s not nessisarily true.