If you want to argue that people mean that more broadly, [2] gives mainly the same result, except that few become center-left.
I would certainly not call anything on the right side of middle in either of those graphics mainstream besides WSJ and Fox. (And frankly, I think it's arguable whether those two are mainstream) So for the US, anyway, it does seem credible that "mainstream media" leans left.
[1] https://www.allsides.com/media-bias/media-bias-chart
[2] https://adfontesmedia.com/static-mbc/?utm_source=HomePage_St...
Going a little further back, NYT sat on the NSA dragnet surveillance story (Risen and Lichtblau) in the 2004 election year when it very well could have cost G.W. Bush the election.
WaPo has been very careful to offer pro Trump editorials all along.
Neither can bring themselves to write "Trump lies".
I put a lot of blame on unpaid internship culture: the national media tends to skew towards people from affluent families because that’s who can afford the early career steps in NYC or DC at way below cost of living.