The NYT's right/center lean has been well documented and expounded upon by others, including some so-called "rich liberals". BBC has also veered right in the age of Brexit.
https://www.vox.com/policy-and-politics/2018/3/15/17113176/new-york-times-opinion-page-conservatism
https://newrepublic.com/article/146828/lefts-war-new-york-times
And even as I'm quoting Vox articles, I'm personally skeptical of Vox's ideological leanings.
The problem is, NYT, on an organizational level, mistakes being "neutral" with "giving both sides a chance to speak", which is the ideological stance of "centrism", and I don't need to go into detail about how leftists feel about centrists. And you might say "don't conservative opinions deserve space in the op-eds?" Sure, but so do marginalized POC voices. How often do you see those? The slant is pretty obvious once you consider the power dynamics involved. Maligned conservative voices deserve pity and empathy and a chance to speak, liberal minority/under-privileged voices are ignored.
Rather than being a neutral institution, they've taken to reflecting the non-commital, non-confrontational politics of the liberal center and that is a problem for me especially in today's media landscape when it's more important than ever that democrats are activated and invested in the future of politics.
I used to like WaPo but I recently changed my opinion about them lately over something that happened, but I don't recall what.
More broadly, news media is supposedly an institution whose moral charge is to be on the frontlines against misinformation and propaganda, and people feel they're not sufficiently committed to that mission. The only reason we have "free speech" and "freedom of press" is to make sure the press can't be taken over to become propaganda, but this doesn't guarantee that media won't elect to become propaganda of their own volition if they see profit in it. It is difficult for me to reconcile the philosophical underpinning of "freedom of press" with the way newspapers bow and scrape for White House access. Make no mistake, NYT benefits as much from Trump's presidency, in terms of profit, as CNN. Trump is a 24/7 generator of news. Trump guarantees there's always something interesting to write articles on.
I disagree with the characterization that the NYT is a "right center" lean. Increasingly, from the more progressive left anything that isn't "leftist progressive" is being categorized as "center right," it isn't. The NYT is center left by American standards. Sure, by say, the standards of the Netherlands or something, it may be center right, but it's an American newspaper and judged by that American standard it represents a coastal, center-left perspective. When other "impartial" organizations try to take a birds-eye look at American media and classify it into left/center/right, like the Pew Charitable Trust for instance or academic studies from the University of Michigan, they almost always classify the NYT and most other mainstream American urban-center newspapers in this left-of-center area. The only places that routinely criticize the NYT as being a right-leaning publication are places on the ideological poles, it's never well researched, it's never exhaustive, and they make that statement to justify a narrative, not because it's factually true. It's unfounded and incorrect.
When it comes to op-eds, I don't necessarily agree with "equal treatment," but the majority of OpEds in the NYT are left of center, about a 8/2 or 7/3 ratio in a given Sunday Times where the Op Eds are much larger. For the daily op-eds it's usually 13:3 left:right. It's just that every time that Jonah Goldberg or Larry Kudlow has an OpEd in the NYT, it ends up making rounds in more progressive circles, while the 6 or 7 mainstream center-left or further progressive left OpEds in that same issue aren't remarkable. Further, I think there's a tendency to take someone like Goldberg -- a neoconservative anti-Trumper who certainly enabled the military adventurism when neoconservatives controlled the Republican party and likely led to a world where Trumpism could take over the GOP -- and convey him as a paragon of the far right, but in American politics, he's not, he's genuinely conservative but he's not far right or populist right. Now, I'm using Goldberg as an example because he's someone who frequently appears on NPR or in the Washington Post or Meet the Press to represent "the congenial conservative." I'm sure that there are other more extreme conservatives who also appear in the NYT OpEd section, but I don't read their OpEds and have little interest in them... Some Goldberg is typically the guy I fall back on. I'm sure there may be some more extreme editorialists that they feature, but I don't read them because I have a brain and can choose for myself. Generally, the readership of the New York Times tends to skew liberal, generally tends to skew more informed, generally tends to skew more educated, and we all have brains and we can decide if someone is stanning for Trump in a particular OpEd page, that we can read the 5 or 6 OpEds that are critical of Trump on that same page instead.
You're right that the NYT takes a non-confrontational approach to reporting the news, but I don't think that news organizations dedicated to reporting should take a confrontational approach to reporting the news. That's why we have modern tabloids like the Daily Beast, the Atlantic, or Slate (and I don't mean 'tabloid' in a negative way, it's actually how the Editor in Chief refers to the latest incantation of the Daily Beast: the modern broadsheet, A respected publication that is willing to be confrontational, do a deep-dive, and call someone out in flowery language). There is a place for confrontation, but there is a greater need for deep investigative journalism, embedded journalists, and people working the daily beat of reporting -- which is something that the Daily Beast cannot afford to do, and so they don't do it. I saw a lot of criticism of, for instance, Maggie Habberman from the NYT for her daily coverage of Trump, as she was assigned to the TRump campaign and is the White House press reporter, and she has surprisingly strong access to a president who is openly hostile to news organizations like the Times. People have said like, "How can you defend Maggie Habberman?" in the past, and it's like, well, I shouldn't have to: She's won two Pulitzer Prizes covering the Trump Administration and campaign; so much of what we have the president on record saying has come from Habberman and her colleagues (most of whom, admittedly, I don't know their names), and so much of the reporting that other outlets like the Daily Beast or what have you can use to tear this administration down comes from the embedded reporters at the NYT, WaPo, and those other outlets that are committed to funding embedded journalists. Today, we've got an over-abundance of confrontation, and over the last 20 years, a perpetually dwindling dearth of non-confrontational committed reporting, reporting facts and reality, instead of appealing to ideology and emotion. Trump has, of course, reversed the trend in the decline of journalism and we're in the midst of this ~4 year window where organizations like the TImes and WaPo can reverse their economic forecast, but that's probably not a longterm trend, and there's a likelihood that national reporting organizations like the NYT and WaPo could go the way of local reporting organizations, and become like the thousands of local newspapers that have had to shutter and close... or be gobbled up by conservative media companies and turned into daily smut. If that happens, our democracy will be in much worse shape than it is today.
I'm open to the argument about how newspaper press can be harmful when covering a candidate like Trump, because when Trump dominates the collective American psyche for a year, they end up writing far more articles about Trump in a given news cycle than other candidates. And, for sure, the NYT profits off the Trump presidency like CNN does, but... so do you, so do I, so does this forum, so does far left Reddit, or whatever our preferred community is. Every day this forum is filled with a dozen fresh threads about Trump, thousands of posts, hundreds of thousands of views of those posts a week. You and I, critics of Trump, are likely to see our posts quoted, and nodded along to. For sure, nobody is paying us for these takes on Trump, but our egos get that little boost anytime we say something about Trump that other people agree with, a little social equity to boost our spirits. Trump is a lightning rod and always has been, and it's tough to criticize a reporting outlet like the NYT for covering Trump and profiting off of that coverage without also criticizing ourselves for being as obsessed with the daily machinations of Donald Trump. If Trump weren't profitable and the NYT couldn't profitably cover him, almost everything that we know about the TRump/Russia relationship, the Trump ORg's shady business dealings, where the Trump family got their money from, the hush payments, and all else, wouldn't be known, because it's those dogged outlets like the NYT, WaPo, or what have you, putting the money into hiring journalists to dig into that and report on that. WHatever our preferred Subreddit, or podcast, or very ideologically polar source, wouldn't have anything to write about or cover without the major investigators putting the funding into investigative journalism.
Where I completely agree with you on is highlighting more people of color. The New York Times is committed to that, and they're aheead of most news organizations in the US, but it's not enough. They publish a yearly report on that, they still only employ something like 30% people of color, but in the last year they've now become a majority woman staffed organization, and among leadership positions, they're nearing 50/50 male:female ration, where just 4 years ago it was closer to 60/40. With race and ethnicity, this is something that needs to be more reflective of New York City as a whole, but the NYT is committed to that, and the numbers have been improving every year since they started tracking it.