Not that firearms makers won't chamber their competitors cartridges (obviously you can buy a Remington in 300WM or a Winchester in 7RM) but some have vested interests in their own cartridges, especially in a smaller niche market like 375 caliber. so I could see some companies protecting "their" cartridge and only chambering their rifles in their cartridges.
Off the top of my head, major North American firearms manufacturers that have invested in the 375 Ruger are Ruger, Mossberg and Savage. I can see Remington holding out as they have their 375 RUM and WBY as they have their 378 WBY and the excellent 375 WBY (not sure if they still produce 375 WBY rifles anymore though?)
However, there is no .375 Winchester Magnum cartridge (just the 375 Win for lever actions) so they possibly will be open to chambering the superior 375 Ruger cartridge. Hard to say though, as Ruger essentially produced nearly perfect rifles to house their NEW KING. Anything Winchester could do would basically be a Ruger copy, other than a stainless 24" barrel rifle perhaps.