This is such a poor article that I would expect the by line to have been one of Vox's frequent contributors rather than their core staff (and Dylan Matthews at that). I certainly believe that that the studies countering the original study are valid, and that the original study is likely flawed but the conclusion drawn is not really pushed back on by the rebuttals.
First off, in straight numbers: if 10% of the population is winning 50% of the time (when there is a difference) then that is direct evidence that their share of influence is greater than their position, and it validates the other study, does not counter it. (Yes, the 0 influence of the other is countered, but winning 50% of the time when you make up a much larger fraction of the electorate is losing badly.)
Second, the methodology is questionable (possibly of the original as well, but certainly of the rebuttals). There are lots of bills that go through congress, and there may be differences of opinion on any of them, but you really want to isolate the ones that one group or another particularly cares about. Next, the top 10% probably isn't really the most relevant group here, since a big fraction are professionals who may have some issues that really focus on them, but when discussing the oligarchy you really want to focus on a much smaller subset, and it is most likely the top 0.01% that is of the most relevance. That is a group that is much more impacted by legislation pertaining to how the economy works. Finally, it is legislation pertaining to how the economy works, not even mainstream issues like taxes and social security that are the problematic issues. Focusing on "economic issues" is not really telling as the category is so broad as to be irrelevant. The way the extremely wealthy influence policy is not the big bill but the little amendment, or exclusion to law. The Trans Pacific Partnership is actually a fair thing to look at, but not as a free trade issue as much as how it is a bill to enforce US Copyright/Patent protections abroad (more than just that but good example).
There is also a bit of a point in there regarding interest groups and how they do get more of what they want, but while the original study separated interest groups into pro-business and pro-public, the rebuttals don't. Since it is through interest groups that wealthy can influence it seems odd for the rebuttal studies to group all of them together and say that the wealthy don't have much influence because influence groups do...which groups do and who controls/supports them is very relevant.
Finally, Dylan is very oddly dismissive of the points made by the original authors. Their points were along the lines above and are quite relevant. Perhaps there is an issue with their original study, but the conclusion is hardly countered by the three rebuttal studies referenced here. Statistics can tell important stores, but they can also be used to hide important truths. If the very wealthy only care deeply about 1% of issues that come up, though they will certainly have opinions on the rest, and that 1% all goes their way, then they got everything they wanted. The rest is noise that hides the important aspect of who dictates legislation.