Both schools of thought are correct.
There are thresholds for acceptable formulas for either to work consistently enough though.
I'd expect a visual representation to look something like a nonlinear sort of graph where X & Y are velocity & weight and the consistent effectiveness falls on the outskirts of of the line drawn.
A different line would need to be drawn for different game and of course there are many other factors at play. This is an oversimplification.
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For example a 223 has a lot of velocity, but the projectile is too light to use that velocity to produce sufficient terminal effects which a hunter may desire for larger game.
On the other side it is known that guys who try and take deer with, say a 44mag or 45LC lever gun, are likely not going get a "drop em in their tracks" kind of shot. Those rounds will kill a deer just as dead, but you may be tracking it further than a 30-06.
Being in FL where I hunt in thick swamps, I don't like the risk of having to track too much for fear of losing an animal. So I don't tend to hunt with heavy slow "pistol" calibers for the above reason. Speed, weight, or both tends to be my friend as I've seen what it does on the game down here.
There are of course exceptions, I shot an excellent FL swamp whitetail in the neck with a 10.3" 223 55gr GD this past season and he dropped in his tracks. The cartridge was able to do enough damage abruptly enough just underneath his neck vertebrae that it shut off his CNS. I wouldn't bet on that happening every time, but it was an impressive show of what even a 55gr projectile moving at ~2500fps "could" do in the right place.
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Forgive the rudimentary explanation. I was a liberal arts major.
Cheers