The 375 H&H is an amazing round but it’s a belted magnum cartridge without a rim. Then 375 Flange however provides nearly identical ballistics from a rimmed case. Without getting too much into the weeds, and only giving the sparknotes version; the rim aids in extraction/ejection. Using a rimless case (like a 375h&H) can and has caused ejection issues/malfunctions because there’s no rim.
Maybe it’s me, but the last thing I’d want to do while hunting dangerous game is take a rimless cartridge, that has been known to have ejection/extraction issues in a double rifle and risk those issues happening during a life or death situation, which is significantly more likely to occur dangerous game hunting vs plains game hunting. Murphy has an uncanny ability to rear his head and I personally like to try and avoid giving additional places or reasons for him to do so.
I’d take a push feed 375 hunting before using a rimless cartridge like a 375H&H in a double rifle for Dangerous Game. Now a scoped 375 H&H Double for bait or plains game would be a lot of fun if I had a bunch of extra money.