I would say it really depends on what size of bore we are talking about?
A .375 is going to have an effective range comparable to a 30-06, while still being able to be loaded down for animals such as deer.
Once you get up over .400, bullet drop starts becoming a problem unless you are putting massive velocities behind those bullets, which takes them out of the category of responsible cartridges for medium sized game. BC becomes a bigger factor when trying to load big caliber bullets at lower velocities.
Whatever a .375 caliber is going to be loaded for, a 30-06 isnt going to do a whole lot better. It is going to do it more efficiently, that is for sure. And a hot, fast load in the 30-06 is going to have a better effective range than even comparable .375 loads without compromising meat at short ranges - but I wouldn't say at a detriment to those other .3xx calibers.
Now if you are shooting inside 200 yards, I would say there is nil to say that any caliber is better than another with the variety of powders and bullets available for any caliber.
But terminal ballistics and killing power all being equal due to loading within those ranges, the .30-06 is going to do it more efficiently, cheaper and lighter.
Oh and if you have ever shot anything smaller than a coyote center mass with a full house .30-06.... you'd know its not THAT versatile...
POOF!
Now if you stay medium bore, there is a definite argument that the .338 is MORE versatile due to its ability to use larger bullets for dangerous game, while still having the ability to load it down for smaller game and maintain some bit of BC with those bullets.