For elk, bigger is always better. Though the little'ns get the job done.
I'd grade the 30 cal 165 class as marginal and pretty much the same for the 338 with anything less than a 225 NAB especially beyond the 250 yard mark.
I know died in the wool elk shooters who limit their weapons to 338, 358 and 375 cals.
But I also know just as many that are regular elk getters with cartridges from 6.5-06 through the 30-06.
However, which ever rifle you are most comfortable with and remember to keep shooting until it's down.