Except that's not what we see.
Wealth does tend to concentrate--some people more value current spending, some people more value investing for the future/for a rainy day. If everyone were immortal you would eventually end up with all the wealth in the hands of the savers.
However, we are not immortal. While fortunes are made they get passed on to those who aren't such savers--in practice they are dissipated in a few generations at most.
This isn't yet completely established in the US, because yours is a young nation and the 'shaking out' is still occurring. But if you look at a society that has had several centuries for the system to operate largely unchecked - for example 19th Century England - you find that the people with all the money are the great-times-15-grandsons of people who were Henry VIIIs drinking buddies. They are rich because they are rich - and it is almost impossible (in the absence of deliberate wealth redistribution) to get rich starting from poverty; and nearly as difficult to get poor starting from great wealth.
Back then, yes. There was a very big difference back then--those people did not have piles of money in the bank. Rather, their wealth was generally in the form of a single asset that produced income. This made it much harder for it to be dissipated over time--not to mention laws specifically designed to keep it from being dissipated.
The world no longer has such factors keeping it from being dissipated. The heirs will tend to spend it and it will be split amongst multiple heirs. That erodes it pretty fast.
Before the industrial revolution, wealth was often in the form of a single asset
class - agricultural land - but rarely a single asset.
Using 19th Century England as an example, the industrial revolution brought great diversity of investments; and with it, approximately
fuck all change in the concentration of wealth - which pretty well explodes your hypothesis.
Concentration of wealth didn't begin to fall significantly even with the introduction of income taxation; Inheritance taxes started to have an impact after WWI, but it took the introduction of the welfare state after WWII to finally break the stranglehold of the aristocracy on the wealth of the nation, and to allow the very poor to achieve a standard of living sufficient to make a long healthy life a realistic goal for a person born into poverty.
Inheritance can dissipate wealth, if two conditions apply - large families need to be typical; and several members of the next generation must typically inherit a sizable fraction of any given estate. The former was true before WWII, but is less true today; the absence of the latter, however, was a defining feature of the aristocracy in England for at least nine centuries. I see no reason to expect that such voluntary dissipation of wealth will ever become the norm; and indeed we observe that it was not normal in the past. Unless the law compels parents to give all offspring equal status in their wills - something that it does not currently do - there is no reason other than pure hope to imagine that wealth concentrations will dissipate naturally over time.
The idea that you can already determine by observation that things 'now' are different from 'back then' with regard to inheritance is crazy - there have only been three or four generations in total since WWII, and the massive shift in inheritance taxation, and the provision of welfare (funded by income tax) to the poor that has occurred since then, will mask any such trends that might exist independent of the effects of inheritance taxes for at least several more generations yet.
Once a person has sufficient inherited wealth, it takes a lot more than being 'Not much of a saver' to dissipate it. It is possible to go from a mere millionaire to penury in one generation. But it is very hard indeed to go from being a
billionaire to penury, without actively trying to lose one's fortune. Even with a fairly poor investment strategy, and a profligate party lifestyle, it is difficult to spend your money faster than it accumulates, once you reach that kind of wealth. And most of the ways that it
is possible, involve transfer of that wealth to
other already wealthy people.
A billionaire might lose his fortune on Wall Street (if he tries hard enough), but none of those losses put his wealth in the hands of the poor - indeed to lose that sort of money requires either an extremely abnormal degree of idiocy, and/or to be the victim of fraud, and neither hedge fund managers, nor fraudsters who bilk billionaires, generally give the proceeds to the poor and destitute. He might even be able to lose his fortune gambling at cards, or on horses, or any number of other gambler's vices - but such losses necessarily go to those who have already sufficient wealth to cover large bets.
An individual wealthy person (or family) might fall on hard times; but rarely does this occur in a way that does anything to significantly reduce the concentration of wealth - the beneficiaries are typically other wealthy people, and/or few in number, and so the overall position remains unchanged.
It is possible for a hyper-rich person to disperse his wealth in ways that do not simply further enrich other hyper-rich individuals; but it is far from common; and to avoid the trend of wealth concentration over the long term, it must not only be common, it must be typical.
Only by imposing progressive taxes, plus sizable inheritance taxes, and
using those tax revenues to benefit the poor, can the trend towards ever greater concentration of wealth be mitigated.
Great Compression had never happened.
