Kudos to the OP on this one - it's a well deceivingly well designed thought experiment.
On the surface, it can be answered by an unsupported subjective experience bias - I've never lived in a state with a shotgun only season, so I've never learned to favor shotguns over rifles. For other folks who have spent time living in those areas, the devil-they-know is the slug gun, so they'll support it over either rifle. The younger crowd is more prone to have experience shooting AR's than revolver cartridge leverguns, so they'll be more likely to shoot from the hip and favor the 223rem. I'd venture folks from states where 22cal CF's have been legal for many years AND the terrain is open, such as Texas, would favor the 223rem as well, whereas guys from back east would subjectively select the short handling, open sighted levergun in 357mag over a 223rem which might leave their deer to run far enough to get lost in the woods... Any of these are valid for an open ended thought experiment, however, answering with an experience bias doesn't really give the game its due.
For what it is, this is really well designed - I hope the OP realized that, rather than stumbling on it accidentally. By throwing short ranges and small deer in the mix as the application, it eliminates the experience bias of guys like myself, who have always lived and hunted in states where bucks average heavier weights than men. This leveled the playing field for a real spin. The 3 cartridge options which he chose are all "challenged," (maybe "marginal" isn't the ideal word for some folks?) and all advantaged in different ways. The .357mag has minimal kinetic energy, but heavy bullets to the bullet momentum is high, so it also has a trajectory like a half court shot, but it can be found in rather accurate rifles. The Slug has plenty of KE, heavy for substantial momentum as well, and also has a rainbow road flight path, but typically the precision is lacking for the sluggers. The 223rem has plenty of KE, but the light bullets drag down the momentum substantially; unlike the others, this one has a flat trajectory and can be had in incredibly precise rifles. What one doesn't have, the other two do, all in a round robin game of have and have not...
Personally, I don't care for any of the 3 options, hence I don't commonly field them, but I have taken deer with all of them - bigger deer than the experiment... I don't favor imprecise hunting firearms, so it rules out shotguns for me. If under 100yrds were the game, I'd pick the 357mag, for the "occasional 200yrds" game, I'd take the 223rem. So for this particular game, I pick the 223rem.