I'm 1,000% for the idea of a statistic that captures a more holistic view of economic improvement.
The problem is, nobody's come up with a single number that feels ideologically neutral, because there's no agreement on what the "right" level of inequality is.
GDP is a sum you can't argue with (and easy to argue that more = better, even if not always true), and likewise unemployment is a percentage you can't argue with (and easy to argue that less = better, again even if not always true).
But even if we separate out GDP by class... it isn't clearly actionable. There's no obvious equivalent of "more = better", because it's clear that wealth increases should be spread out, but it's not clear they should be spread out equally.
The puzzle piece that seems to be lacking is some kind of rigorous justification for what the right level of inequality is in a society. But no political scientist, economist or philosopher has yet developed/proved that in any way that people generally accept.
We can't measure the economy all "right" until we collectively decide what "right" even is.