Actually you could use taxes because GDP formula using income approach can be also expressed as value added at basic prices + taxes less subsidies. However, you can’t do it at the same time when using the approach in your question as you would be double counting.
$$GDP= w+i + r+ \pi + o$$
Where $w$ are wages $i$ interest income $r$ return on capital $\pi$ profits and $o$ something you forgot in your question adjustments for things like net foreign factor income, capital consumption and revenue from some gov enterprise (not tax revenue) etc.
You would be double counting if you would include taxes, because taxes are subtracted from these. If you earn 100e and government taxes 10e then economy does not suddenly have 110e. The 10e is just transfer.
If government invests the money in some productive way it will show up as an gov revenue from those productive activities.
The value of public goods like army is already reflected in the income approach as soldiers earn wages, military contractors profits etc. The same holds for any other public good. Let’s say you build a road there are wages to be paid, contractors get profit etc.
Also, when government engages in pure transfers - for example redistribution this can’t show up on national accounts as it does not create more output in the economy, it’s just juggling output around.
However, this is not because government is viewed as parasitical. If you are Ralwsian or Utilitarian transfers from rich to poor are moral imperative. But economics is not moral philosophy it’s a science and GDP is a measure of output that is grounded in real production and fact of the matter is that transfers don’t create any additional output by themselves.