I have made quite a few of both. The 338-06 was a bit more popular here in Wyoming then the 35 Whelen, but just a bit. 35 Whelen supposedly had factory ammo available but we'd seldom see it and I don't think I ever met a 35 Whelen fan that didn't hand load anyway. I have never heard a single complaint about either one that could be blamed on the cartridge. A few reports have come to me and one time I saw a ba dfailure when a slammish deer was shot, but in all cases the blame could be placed on a bad bullet for the job. (In the case I just named however the bullet that blew up was a 338, not a .358)
I can see a reason why there are some bullets in .358" size thast are not right for the 35 Whelen. That being the existence of the 35 Remington and it's popularity. Bullets with thin jackets, made to expand at impact velocities of 1900 FPS and all the way down to 1300 fps do not hold up well when shot at the much higher speeds available in the 35 Whelen.

BUT I see NO reason when ANY 338 bullet should come apart easily. Yet I have personally seen them fired from the 338-06 and had them break up just like a big varmint bullet. The 225 grain Hornady did this and so did the 250 grain Round nose (that one was a shock to me, but I was there and saw the deer shot, and I was the one that gutted it and butchered it and that is the way it happened. The bullet went only 6" deep, destroying some of the spine and we found the flattened empty jacket laying against the spine of the deer. My friend Randy who shot the deer had to kill it with a 22 revolver because it was crippled, but still alive when we got down to it.

I have seen the same (slightly worse) performance from one kill I made with my 9.3X74R with a Speer 270 grain bullet on a 120 pound white tail too.
So bullets should hold together at least 50% on any big game rifle in my opinion, but if you use a bullet that doesn't break up badly, any of the mid bore guns work wonderfully.