The default HDI formula devised by the UN is a mix of GNI per Capita ADJUSTED TO THE PRICE LEVEL OF THE COUNTRY; expected years of schooling of children at school-entry age and mean years of schooling of the adult population; and life expectancy.
Literally in any Developmental Econ 101 class HDI is pointed to as the primary metric to compare regions.
Of course there are valid criticisms with weightages in HDI, but no one and literally no Economist or Policymaker worth their salt has straight ignored it.
GDP PPP per Capita and Gini are critically important as well, and are factored into modified HDI indexes as well, for example Inequality HDIs which add the GINI, and the post-tax Gini Ineqaulity HDI because the naive Gini formula didn't take taxation and income transfers into account.