Very often it's because of two things:
Faster powders are used in many cast bullet loads, necessitating lower velocities, and,
Accuracy is often better with lead bullets at lower speeds, depending upon the rifle, barrel twist rate and bullet fit and hardness.
Too much pressure with a relatively soft bullet intended to expand on game can be a problem indeed. Any severe deformation under the stresses of high pressure firing can distort it to some degree and make a somewhat less accurate load. Obturation of a bullet is not always a good thing. It can also unbalance the bullet. But the whole obturation thing is problematic, and often doesn't occur even though calculations say it should be happening. If a soft bullet isn't shooting, harden it up and see what happens.
I have no great difficulty duplicating factory load speeds with quite sufficient accuracy for hunting, but usually lower speeds mean better accuracy for me. It's the much safer way to bet. The question is whether the accuracy difference means anything.
Shooting a 30-30, 35 Remington etc. with cast bullets at no loss of velocity compared to factory loads and only a relatively insignificant accuracy loss compared to lower velocity cast loads is so commonly done that Marlin shooters regard it as a rather mundane task.
I shoot a 225-230 grain (depending upon lead source and lubricant used) gascheck bullet out of relatively soft 12 BHN wheelweights at 2050 fps out of my 35 Remington with a number of different powders. Accuracy is better slower or with a harder bullet, but the rifle shoots very well enough as is. The idea with the straight wheelweights is to allow a fairly rapid expansion for deer hunting. If I was black bear hunting over bait the bullet would be better at 16 to 27.
And my SKS likes cast bullets just fine, too so your idea for the 7.62 is a good one. Lee makes a good 160 for it and it does well at just under 2000 fps in my gun. The 7.62X39 makes a good cast bullet cartridge and is plenty good enough for deer. It will shoot the cast loads as well as any jacketed. Filing a small meplat on the 160 Lee makes it a bit better.
Leading really isn't the problem with lead bullets that it is made out to be. A velocities common to the levergun rounds any halfway decent lubricant (50-50, etc.) will hold leading at bay with a gascheck bullet very easily. I don't clean my cast bullet shooting levers very much because they don't need it very much.