Like buybacks vs regular dividends, the main differences have to do with their taxability.
From what I can tell, the main complaint in the Atlantic article is that CEOs time buyback announcements to coincide with their stock sales. The first example they give is the Home Depot announcing a buy back (apparently in the Feb ‘18 earnings call), and then selling stock after the insider lock up window opened. In all likelihood, that sale was scheduled well ahead of time (execs have to do this to avoid insider trading charges).
So, the controversy seems to reduce to Home Depot having a strong quarter and buying back stock, making money for shareholders, including the CEO.
Note that if they’d issued dividends, and they pay retained dividends on unvested RSUs, the net effect would be exactly the same (except taxes).