I have seen several people in similar positions go for the 375, and then for a while it's a thing but after some time it's back to the safe. I understand why, my 308 is cheap to shoot, kills all normal antelope equally dead, and is actually fun to practice with.

If I had a 308 or 30-06 I would just skip the 375 all together, because the want will soon grow to I actually wanted a real big bore, now stuck with a medium bore. I understand practice is not cheap, but impala makes cheap bullets for all calibers, and brass goes a long way with proper reloading. Secondly most people just practice with their range rifles, they don't do range sessions with big bores, for reasons mentioned. Sure go hunt with it, even take it for every hunt, just remember most folk don't actually hunt that much, they mostly just shoot for most part of the year.
So, DONT let the hunting requirements blind you into buying something you don't want, go look at your notes and figure out how many animals you hunted the last five or ten years, for most people it's not even one packet of hunting bullets in quantity.

In that light, I would just go for something like a 416 and be done with it.