Sorry, what? Bertelsmann is a lot of things, but left-leaning ain't it. They are true neo-con/neo-liberal free-market fetishists. They are only left-leaning in so far as they are extreme-right-rejecting, with their Bertelsmann Foundation acting as their vehicle for lobbying, e.g. deregulation of the television market (where Bertelsmann is a big player with RTL). They are a lot like Alex Springer in that regard, as both are very "free market" and rather conservative (unless there is money to be made) and against too much "socialism" and against nazis.
They are also a direct Alex Springer competitor outside of television (where Springer thus far only has a minor presence after they repeatedly failed to take over RTLs competitor Sat1Prosieben) as Bertelsmann owns Gruner+Jahr (magazines, tabloids and newspapers), Penguin Random House (largest book publisher in the world) and additionally companies like BMG (music publisher), and the Arvato you mentioned (who not only do "content moderation" but credit checks, logistics and all kinds of other services).
Alex Springer and Bild, and Julian Reichelt at it's helm, is powerful, but a lot less so than a lot of people would want you to believe, and a lot less than they used to be. But I'd agree that Bertelsmann is more powerful/influential in Europe and Germany than Springer now.