In Piketty's credit, he did make this easy. Providing data for reproducibility and replicability is relatively new in economics. There are, however, two kinds of claims here. One is that mistakes were made, which is grounds for schadenfreude for those (like me!) who don't like the slight jumps in reasoning from "inequality is rising" to "something must be done about inequality" to "this is what must be done". For all…
"But secondly, it is claimed that he cherry-picks. Economists (we) do that. A lot. And it has to be called out." The total effect of his cherry picking can't even be discerned from this article. With a lot more work (more than a single blogger probably has time for), it might be shown to be much worse. For example, Piketty chose the U.K., France and Sweden for his list of European countries (averaging them all into "…
https://en.wikipedia.org/wiki/Simpson%27s_paradox
See in particular the Berkeley Gender Bias case if you don't want to hack the math apart.
The specific way you slice and dice data matters.