Why not 35 caliber and use pistol bullets?
Are you asking Spook about his .375, or just a general question for the forum?
If it's a general question - my 358 Herrett can use pistol bullets, as well as most other 35 cal rifles. The problem is in the construction of most pistol bullets, and how they behave at rifle velocity. Some will actually come apart in the bore, as evidenced by severe lead and copper fouling. (When you get lead fouling from a jacketed bullet, you know something aint right!) Others make it to the target just fine, but blow apart on impact, which can be great if you want a varmint bullet but useless for hunting anything bigger than a coyote.
I used to load 38 Spl 158gr JHP bullets in a 35 Remington lever gun for trimming tree tops in the valley in front of the house (don't worry, safe shooting direction and all that). They were pretty impressive on impact fragmentation; two or three rounds would cut down a tree or branch the size of a normal man's leg; far more damage from those than any varmint rifle.
Last weekend I worked up a load for a friend's 358 H (one of my barrels) in a 10.5" AR with the 140gr Hornady FTX bullet, the one intended for 357 lever action rifles. A relatively mild load to just cycle the rifle was ~2250 fps, and the bullets fragmented on impact as expected. I wouldn't want to rely on that for any penetration. At 1600-1800 fps they seem to work pretty well though, and cycle my subsonic-only barrel with the big gas port; that load is basically a duplicate of 357 lever gun ballistics and is very mild to shoot.