A bullet kills game in one of two ways. It can directly destroy the brain or central nervous system that in turn controls the organs that provide oxygen to the brain, or it can destroy the organs themselves. Brain shots on game are very specialized, so mostly big game hunters attempt to disrupt the heart or lungs of the game animal. The wound volume produced is proportional to the expanded diameter of the bullet combined with that bullets ability to penetrate (momentum). The larger the wound volume, the faster the oxygenated blood is prevented from reaching the brain, and the faster the animal succumbs to the wound. Shock may in fact knock an animal down, if that animal is in fact susceptible to shock and not all are, but this has nothing to do with killing the animal.
There are other considerations though when choosing an appropriate rifle cartridge. One consideration is that when used for varmint hunting, the importance to minimize the ricochet potential of the bullet. If this matters, then a light high velocity bullet like a 55 gr Nosler BT from a .243 at nearly 4000 fps would be a better choice than the lightest 7mm bullet available.
My preference is a minimum bore diameter of 6.5 for big game, but this choice is aimed at a general use big game rifle used on all non-dangerous big game. A rifle that is used as both a varmint rifle and a deer rifle fills a different niche, and as such the bullet availability of the .243 makes it a good choice.