I've a custom 338-06.
Overall I like the round. It is a handload only option for me though. The Nosler ammo you are looking at is around $80/ box of 20. With the chamberings you own it would fill a niche. You get a little extra bullet weight and .03" larger diameter over a 30-06, but real world performance is not really any better. There have been several studies/tests done over the years comparing the round. The 30-06 with heavy bullets actually outperformes it and the 35 Whelen on larger game by a small margin.
The 338-06 load you show is a 225 Gr accubond @ 2600 fps. A 30-06 will shoot a 200 gr accubond @2700 fps. A 300 mag at 2900 fps. They will shoot flatter, and because of the better sectional density will penetrate deeper. The extra .03" bullet diameter will never be noticed.
Comparing it to the magnums you may be able to save a small amount of weight. but a couple oz. at best. My 300 mag weighs the same as the 338-06. Recoil is the same too. There is no free lunch. Start shooting heavy 225+ gr bullets from a 338-06 or 35 Whelen and you get magnum recoil even though it does not have "magnum" on the headstamp. The greatest advantage is that you get 4-5 rounds in the magazine compared to a magnums 2-3 rounds.
From a purely practical standpoint a lightweight 30-06 would be a better choice. Ammo is easier to find, much cheaper and it is more versatile. To be honest your 308, with proper loads would work for what you want to hunt.
But if you just want a 338-06 it is a very viable option. It shoots flat enough to be effective at longer ranges, it certainly has the power for anything in North America and if you handload there are lots of bullet options from 160 gr up to 300 gr. And the main reason I keep mine, you are hunting with something a little unique and different than every other hunter on the mountain without sacraficing anything. Just keep in mind you aren't getting any real advantages either.