The first point I am trying to make is that it should've been obvious, and that aside from giving an actual name to the entity through which cash went from Kremlin to Milner, the NYTimes is not news. If you were concerned then, good. If you were not concerned then, there's no reason to be concerned now, as no new facts came to light through this NYT article.
On the second point: the source of the money should not be cause of concern if it's "just money."
If it's money and a very strong suggestion that you partner with a (obvious, even if explicitly unstated) foreign, state-controlled entity, then it's a cause for concern. If it's just money, and then when you want to partner with a foreign, state-controlled entity, that same money gives you an introduction to the foreign, state-controlled entity, then it's not a cause for concern, because that's the point of business relationships and investments.
The question of Russian influence is really a separate question, in my mind, and I think that's what bugs me about this entire thing the most. The implication somewhere here is that the leaders of Facebook and Twitter, and other startups that take Milner money were somehow duped, and allowed Russia to walk free in their systems and influence the election. I will be very surprised if that's true; Zuckerberg and Dorsey are too smart for that.
What seems obvious to me is that the systems they created are giant, targeted megaphones, by design. They were successfully used by the Obama campaign, and everyone was very happy for democracy then. Then a foreign state decided that they were going to give Trump help on these megaphones, and now everyone is very unhappy for democracy. Would we have this much hand-wringing if we discovered that Saudi Arabia was helping Hilary?