A longish reply - back in the day, I "only" had a 308 Win for deer - I took many dozens deer with that and 165 grain bullets over the decades. My Dad used a 30-06 - about no difference - my hand loads versus his factory loads - until 200 grain bullets and heavier - I could not keep up with the 308 Win - although I got my first elk with that 308 Win and 165 grain bullets. His last elk was with 30-06 and 165 grain bullets. I bought a Savage 112V in 22-250 for myself, and a Remington 788 in 243 Win for my wife to hunt with - I suppose nit pickers will find a difference. I shot at many gophers with 308 Win and 110 grain hollow points - even hit some, I think. Our son (now 46) got his first whitetail with his Mom's .243 at about age 12 or 14. Our daughter got her first whitetail with it as well, several years later. No flies on a 243 Win. Or a 22-250. Or a 308 Win. My next "carefully considered" purchase was a 338 Win Mag - I got two more elk with that one - those shots were circa 100 yards and about 12 feet - so about anything would have worked for those shots - but I lugged that 338 for many miles in Saskatchewan Spruce swamps.
Currently, when I am bouncing along back roads here in Manitoba, chances good the little 16" barrel 243 Ruger 77 is beside me on the truck seat. You are like looking at a "picket" fence - one cartridge is going to overlap a lot with the next one - will be "nit picking" differences. But no doubt a .223 Rem and a 375 H&H are going to have different roles - but there are myriad of "pickets" in between those - is not cut and dried - some will talk 22-250, or 225 Winchester, some will talk .257 Roberts or 25-06 - some will talk 7x61 Sharpe and Hart or 9.3x62. Is up to you what you need or want - and then can look at 7mm-300 Weatherby, 416 Remington Magnum, 458 Win Magnum or .505 Gibbs - just more pickets, likely outside what you have already have ... Is about no end to it.
You have two of many cartridges that I have never owned or used - 7mm-08 and 270 Win. I have never used a 7mm Remington Magnum either. Not against them, at all - is just that there are many "pickets" really close to those.
If, today, I was forced to go back to "one" - it would probably be back to that 308 Win - shoot at gophers, coyotes, deer and gongs. Or the 7x57. Or similar.