The idea of using log-income as a proxy for utility is not new. It is a standard measure in the economics of taxation. Yes it is different than measuring consumption. It is an improvement over GDP in the sense that it measures how the economy is serving the population. Some consider this to be a laudable goal.

The economy is serving the population when it enables consumption. Income is shifting numbers around in banks, consumption is delicious mac&cheese.

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.

If you include wealth transfers such as inheritance in consumption, and you include mortgage spending, then log-consumption can work fine too. The only problem is that it is more difficult to measure because you would need to track either either consumption per-person or both income and savings per-person.

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.