That is exactly what state game and fish agencies do, very successfully, with basically everything else (black bear, mountain lion, wolves, elk, whitetail deer, mule deer, moose, sheep, goats, turkey, grouse, doves, pheasants, quail, ducks, geese, cranes, fish, squirrels, rabbits, etc., etc.) They monitor population numbers and habitat conditions, and set quotas as appropriate to maintain ecosystem balance.
Hunters have been able to shoot big grizzly/brown bears uninterrupted in Canada and Alaska this entire time, grizz hunting is not some unobtainable thing. IMO this isn't about pent up demand for grizzly hunting. I don't know the ranching situation around Yellowstone very well, but I do know the Park Service in Yellowstone artificially increased the numbers of bears by feeding them garbage for the better half of the 20th century, and then stopped in the 1970's, which resulted in a huge reduction in population numbers, because the habitat couldn't support that many bears. It may have led directly/indirectly to the ESA listing, I can't remember the timing there. Hunters didn't create this mess, we all did, and hunters are funding the lion's share of the restoration efforts.