This is not the general case. This is not about some arbitrary private organization, but rather one of the two major party organizations that select candidates for the general election.
To review, the U.S. election system for presidents is single-winner, which game-theoretically pushes the general election toward a two-candidate race. Those two dominant candidates are selected by "private" political organizations called the DNC and RNC.
Was it legal for them to do what they did? I guess so, but I'm not a lawyer much less an election law lawyer. Does a system in which the DNC can secretly (until it gets leaked) show favoritism still represent democracy of any kind? I don't think so. It's cronyism or oligarchy or something more akin to that than to representative democracy.
So in the Democratic primary we had the DNC (legally, presumably) influencing primary voters to favor Hillary. In the general election, we had Russia, to some extent, influencing voters to favor Trump (or at least hate Hillary enough to stay home). Everyone goes through the roof that a small group of people from another country would influence the general election, but little to no concern domestically that a tiny group of people were influencing the Dem presidential primary results.
If you would prefer the term "influence" instead of "rig" in reference to what the DNC did for Hillary vs Bernie, I understand that. But none less than the NYTimes declared (via an editorial by Krugman) that Russia "rigged" the 2016 election when what they really mean is that Russia appeared to have influenced voters, which is the same as what the DNC did to Democratic Party primary voters.
If the RNC secretly did the same thing, I'd be complaining about them too, but there's a lack of evidence that they did, and even if they tried, it wasn't successful. As you pointed out, there were some public efforts to bias (against Trump), but they were too incompetent to be successful.
The RNC and DNC are organizations intended to help their parties' voters pick among several primary candidates. If the organizations' leadership is not neutral (or doesn't try hard to be neutral) but pretends to be, primary voters risk being seriously misled. As I said before, the whole notion that such a system has anything to do with democracy is a sham. So this past election was a sham candidate vs an awful candidate who was at least honestly selected by general party sentiment.