No there are several flaws in your argument:
- Starting with the most obvious problem. Just because something will be eventually owned by someone does not mean it is correct to say that the thing is owned by that entity already. This is equivalent of stating: "My house will be once inherited by my posterity so I don't own it but they do." This is flawed from economic perspective. Ownership (property right) is defined as (see Mankiw Principles of Economics 8ed pp 11):
the ability of an individual to own and exercise control over scarce
The definition might be slightly different in different strands of research, but generally ownership in economic sense is determined by who has exercise/control over something. Until government actually seizes the ownership of the thing it from economic perspective does not own it. Furthermore, I am not a lawyer but I am going on a limb here to say I doubt this would even hold in legal sense, so the assertion is flawed prima facie.
Next, typically government does not just hoard resources it gets (in your example currency), but also spends it. Since you are building theoretical model you are free to make assumption that government never spends its resources but that assumption does not hold in real world so you cant automatically claim conclusions from your model carry there. Models do not have to be 100% realistic (what even more they should not even attempt to do that - they are models) but in your case assumption of 0 government spending is the primary assumption making the argument, but empirically it is clearly invalid - that is serious flaw in this case.
Even if we forget about the point 1 and 2, you are making claims that go way beyond the model you use. In your model government taxes currency. It would be true that if there is fixed amount of currency and government taxes it at each point when it is through spending turned into income, then eventually in the limit government would own all currency.
However, currency, or money more broadly, in itself is neither income nor wealth - it is way how we measure income or wealth. Income is generated by production. If you produce two widgets, value those widgets are your income. Money that you will get from selling those widgets measure (and store) value of those widgets. If government would tax away all currency people would just revert to barter economy where they would exchange goods and services for another goods and services and those would be peoples' incomes. Government would own all currency but not peoples' incomes.
Now this is not just an attempt at 'gotcha' - this is important distinction. If government would be taxing goods and services in-kind, then government might never get hold of all of them as people can always produce more and more goods and services ad infinitum.
In addition, you have to be clear with your terminology. Income is not wealth. Wealth is defined as net value of your total assets which are accumulated by saving. If you spend your income it is not wealth and consequently saying: " most wealth in circulation belongs to the goverment it circulates in, as long as it keeps circulating?" Is incorrect. Because what is circulating in your example is income not wealth.
Lastly, the same caveat that I mentioned about income applies to wealth. Wealth is not simply money. Person with no money but ownership of billion dollar company has more wealth than a person with full wallet of banknotes.