Op has a 6.5x55, which could do anything a 243 could do and better. The gap seems above 3006, with the 45-70 covering close bush/defense.
While the 375 h&h can do anything a 300 or 338 can do, it really shines at close to moderate distance. Your missing anything at extended distances needing energy. There is also a giant gap in ammo prices for anything north of 300 magnums. Would imagine it's a big jump to 338 mag(been awhile since I've bought factory ammo), but 375 ammo could be over $6/shot, before this recent insanity.
You're right, I am lacking something with a little more oomph at range, however for the type of hunting that I do and the terrain I hunt leads me to prefer the shorter range thumpers.
I will realistically never take a shot over 300m with most being well inside of 200m. My attraction towards the .375 is mostly due to nostalgia and it's history but also to own a gun in something like .375H&H would make for a real centerpiece in my collection. I have been browsing 375 ammo prices lately and it seems like they range between $4.25-$8/round. This won't exactly be a weekend plinker at those prices. I don't see myself buying anything bigger than .375 (like 416 rigby etc.) since ammo prices get truly insane at that level.
I would eventually like to partake in long range target shooting in the range of 1000m. That's where I have been considering one of the more modern 6.5 cartridges or possibly a 30cal equivalent.
From my current collection:
1. 22LR - small game hunting, gravel pit plinking and soon to be long range .22LR bolt action.
2. .223/5.56 - Black rifle plinking, could be used on varmints.
3. 9mm - range toy
4. .38/.357 - range toy, could be used on varmints, small deer and hopefully one day wild boar.
5. 7.62x39 - range toy, could be used for a close range deer/boar rifle.
6. 6.5x55 - Medium-Large game hunting rifle (close range)
7. .308 - Medium-Large game hunting rifle (close range)
8. 30-06 - Medium-Large game hunting rifle (Intermediate range)
9. 45-70 - Large game and backpacker rifle (close range)
10. 12 gauge (grouse, clays, deer, plinking).
The .375 will slot in around the .30-06, hunting game within the same ranges but with much more power and more confidence if I ever find myself hunting in brown bear territory.
Long range hunting and long range target cartridges are the two main areas that I think I'm currently missing out on.