.223-don't even think about it. Every other cartridge you named can kill elk.

.243-necked down from a .308 case, not ideal compared to other calibers you named.

.308-a little less powerful as a 30-06, think about a 7mm-08 if you want a short action based on a .308 case.

.270-necked down from a .30-06 case. Higher B.C. bullet shoots flatter than parent cartridge.

.30-06-shoots heavier bullets than a .270. A 35 whelen is a beast necked up from a .30-06 case.

7MM Rem Mag-heavy hitter shoots very flat. My favorite magnum cartridge.

300 Win Mag- overkill on deer, expect recoil.

I would recommend a .270 based on your criteria.

As you being a new hunter I will add this. Most shooters will get a flinch if subject to too much recoil. It will ruin your potential to become a good marksman. Staying under 15 pounds of recoil energy is a good gauge for easy shooting. The .270 would put you a tad above that mark. Unless you get some ultra- lightweight rifle that is.