A little experience with one of the old Remington 660s. What a freaking shoulder breaker out of that particular gun.
There are several problems with the .350 Remington that doomed it to failure...
1. The belt. Belted magnums just aren't necessary. I wish the gun companies would figure that out. H&H developed the belted mag as a way of controlling headspace on bolt-gun rounds with long, sloping shoulders. The belt's usefulness ended as soon as American companies started making ammo with nitrocellulose powder as opposed to British cordite.
2. Traditionally chambered in rifles with short actions, meaning that heavier bullets have to be seated far back into the case, which really chews up case capacity. While it does well with lighter bullets, the .350 is a good choice as a "thumper" in bear country or for other large game at close ranges. Better abilities with heavier bullets in an action that's a little longer than short would be a good thing.
3. It's a .35 caliber. The only .35 rifle caliber to ever truly make it in this country is the .35 Remington. Numerous other, better, .35s have failed, such as the .358 Winchester, and the .35 Whelen isn't exactly vibrant, either.
Those are the problems as I see it.
Even though the round failed commercially the first time out, as did its stable mate, the 6.5 Remington Magnum, I really think the .350 deserves another chance, and I really hope that the shooting public learns about, and learns to love, the .350.
It is a good little cartridge.