This is true of GDP as well, since GDP only includes final goods. I.e. if a farmer produces wheat, and the wheat is then used to produce bread, the farmer -> baker transaction is excluded from GDP. Only the baker -> consumer transaction counts towards GDP.
If you want to try and measure utility, your metric should be based on consumption - e.g. mean(log(after tax consumption)). Income isn't utility, it's only potential utility.
If we used income, your metric would unfairly penalize volatile income. I.e., a person with a stable income of $50k/year would be considered 70% better than a person who earns $100k then $0k (saving $50k in the first year and spending it in the second while having an identical lifestyle.
The distinction between consumption and income is not significant for a large fraction of the population, and if you are going to track just one number, it really shouldn't be consumption. If you do that, you have no hope of tracking changes in social structures or inequality or wealth accumulation. You would have to track wealth or income too.
Addressing your concern over volatile income is not that important and could compromise the usefulness of the metric. If you propose using GDP or any other linear function instead (rather a logarithm or power-law) then you cannot approximate utility and your number will be insensitive to changes in inequality. If incomes go up for the bottom half by 10%, then GDP would not change by much, though many would be better off. Income taxes are already calculated annually to smooth out these income fluctuations for seasonal workers.
But you wanted to measure 'average "utility"', not "social structures". Utility comes from consumption, not income. You are right that tracking consumption would fail to be a proxy for income inequality (as you now seem to want it to be) because consumption inequality is vastly lower than income inequality.
At this point, your metric sounds less like an improvement on GDP, and more like just some other random thing you want to track.
That said, you criticized log(consumption) above because it doesn't measure "social structures" and "inequality". Those things are pretty explicitly not how the economy is serving the population.
Income is easily measured for individuals, and aggregate consumption is easily measured by merchants. This is why we have a progressive income tax and not progressive consumption taxes. It could be done, it is just more difficult.
You can repeat as often as you wish that "inequality" is not relevant, but you are wrong. Utility is increased more if $1 is earned/spent by a poor person than a rich person, ipso facto inequality is bad for utility, all else being equal. My reference to the economy serving the population, is just a reference to utility, as opposed to any other linear measure such as GDP.
The most telling one, for me personally, is the human condition of the bottom 20% of the residents.