Firstly inequality growth is mainly an American phenomenon, it's not really a factor in the developed world generally.
The problem is the Pickety/Zucman analysis didn't fully take into account taxes and transfers, so it overestimated 1%-er incomes and underestimated low wage earner benefits. Within the US, CHIP and Medicaid expansion have made a huge difference. After all it's hard to argue Medicaid expansion has no effect on inequality.
Another major area of anomaly is that the P/Z analysis assumed income from pass-through businesses was capital gains. It turns out, this includes a lot of doctors, lawyers, etc whose income comes form these kinds of business structures. That made it look like their income was capital gains from investments in businesses, where in fact it's really labour income. This is not a small discrepancy, there are a lot of these sorts of people and they earn a lot. Correct for these sorts of things and the inequality largely evaporates.
Anyway, it's a messy problem with a million details and there are still distortions in the economy that disproportionately benefit the wealthy and always have done.
I disagree tax evasion is such a big factor, it's a problem but a relatively constant one. I think the main problems are mostly structural distortions in the tax system that encourage passive incomes. Income from rents and inheritance are significant issues. The drastic rise in property prices across the developed world is a huge problem that disproportionately hurts low income groups.