the main point of PPP is to use GDP per capita as a typical income.
Why do you think so? As far as I am concerned, the main point of PPP is to adjust the FX rates to make comparisons (of incomes, costs, living standards, etc.) between countries more meaningful.
PPP has nothing to do with the relationship between GDP per capita and personal (or household) income.
Yes, PPP is logically independent of GDP, but PPP GDP per capita is quite popular, probably the most common use of PPP.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.