Philosophies and techniques change over time. I personally find the discussion interesting, especially since i have owned both a 44mag lever gun, and 223 rifles.
There are several key differences here imho. But i feel the over all concept comes down to fast and light vs slow and heavy.
44 mag should not be under estimated especially out of a carbine. It has lots of momentum and makes a big hole to hit hard. But it sacrifices trajectory to do it.
223 can hit hard as well. But it is much more dependent on velocity and bullet design to do it. It does however have a much better trajectory and loses velocity at a much slower rate.
I cant say as i have ever heard anyone reccomend 233 for large and or dangerous game. However 44 mag would be more than adequate imho, given a reasonable distance. And im talking a traditional heart/lung, not the 22lr to the eye philosophy. Thats a whole nother can of worms.