I’ve shot 7-08 for several years now. In all honesty all the calibers will do the job. If you don’t reload or don’t want to reload then stick with .270 or .308 for the larger uk species, or the .243 for the smaller.
There are several reasons why I’m sticking with the 7mm-08 (I’ve just acquired another in stainless synthetic)
It’s a lot softer shooting than the .308 or .270, this in weight for weight rifles gives several advantages in my experience. In lighter weight rifles less recoil means less jump, nine times out of ten you can see the bullet strike on the beast and it go down, which I have found not to be the case with .308 or .270. If you see the strike and you know it’s well placed, all’s good, if you don’t know if you’ve hit the damn thing or not and it runs off, well to put not too fine a point on it you’ve got a problem. Also Less jump means more chance at a second shot at another beast. Plus, it’s generaly accepted that lower recoil means greater accuracy.
It’s noticeably quieter as well which is a considerable advantage.
All in all in 7-08 you can have a light weight rifle which jumps less, is quieter, accurate, is easy to reload for, with a good selection of both powders and bullets which will give you similar performance to the .270 and the .308. If you feel that you’re getting to much meat damage, change the velocity or the bullet.
The disadvantages are that factory ammunition is difficult to get compared to the others and quite a bit more expensive.
Possibly the only caliber that I’d like to try over the 7-08 would be .260 Remington.

Target 4 is first try of factory federal premium 7-08 140gn in Sako 75 with a standard synthetic stock @ 100m
Target 3 was two rounds out of the same rifle with 139gn sst’s using Rs52 which had been loaded for my old 7-08, this combination had to be knocked back as it was easily achieving in exces of 3,000fps out of a 20” barrel.
BTW they’re 1/2” squares