They're not. When The Economist agrees you have a point then you're pretty mainstream. True, some economists are nitpicking on his book because they don't like the conclusion, but the problem Picketty has identified is rarely denied.
From the abstract:
In this paper, I build a set of Panel SVAR models to check if inequality and capital share in the national income move up as the r-g gap grows. Using a sample of 19 advanced economies spanning over 30 years, I find no empirical evidence that dynamics move in the way Piketty suggests. Results are robust to several alternative estimates of r-g.
First, he made a lot of methodological errors, like using tax data as a proxy for income and then not adjusting for a change in tax treatment between one year and the next in the 1980s.
Second, more fundamentally, he's computing inequality without accounting for taxes and transfer payments, i.e. the things that we do to ameliorate inequality, e.g. food stamps, the EITC, and the entire Social Security, Medicare, Medicaid complex. I seem to recall that employer-paid benefits which are excluded from tax, like health insurance benefits, are also excluded.
At the high-level, in terms of big theoretical flaws, he posits this broad "when rate of return on capital exceeds the growth of the economy the rich just get richer" theory which is fine until the rich person dies, gives most of his assets away to charity, loses half of what's left to the death tax, and has more than one heir. People sometimes joke about how economists make a bunch of ridiculous assumptions in their models, and "people live for ever" is actually one of them, and it actually makes sense most of the time (because it usually doesn't affect what you're modelling), but this is the case where it affects what you're modelling.
This all started in 2014, so some of it may have been improved or corrected since then, but Picketty's popularity is still independent of any actual measure of facts or truths in the world. It's related to confirmation bias of the listener. This is why he is used in politics so much. (This isn't special or specially bad; it's why anything is, really. But see it for what it is and look to science rather than politicians for a fight on the facts.)
Regarding your "big theoretical flaws" on intergenerational wealth dynamics, here's an entire chapter by Piketty and Zucman on the subject, confirming that their central hypothesis holds when taking inheritance into account: http://gabriel-zucman.eu/files/PikettyZucman2015.pdf
It seems to me that it's quite a crass methodological error to base an analysis on income distribution on ignoring how income is distributed.
No doubt in 5-10 years when the fashionable economics theory du jour concerns debt or housing or tax incidence or technological progress - and large sections of the economics profession do respond - the author will be amongst those chastised for "ignoring" such pressing issues by not abandoning his interest in researching inequality to write papers focusing on Fashionable New Theory instead.
There are definitely some economists who disagree with him, but if you go into any academic field and you'll find plenty of people disagreeing with the field's luminaries.