I don't have a lot of recoil tolerance, so I gravitate towards the typical youth cartridges and other modest recoiling options when I can get away with it. If we are only talking about 100 yards or so of range, I would point you toward 35 Rem as being a LOT more capable than 357 with very modest recoil. Since both the guns and ammo/components are increasingly rare, I'd suggest 350 Legend. Not a classic, but readily available, cheap, and you can even load cast or download in a bolt or single shot. If you insist on a legacy cratridge that is more available, 30-30 is an otion, but I suspect it is going the way of the dodo over time and I was quite surprised with the aount of muzzle blast an otherwise low recoil cartridge has when my buddy was shooting his last summer.
If you were thinking about longer ranges, I would suggest a 6.5 Grendel. This is what I will hunt to the age of infirmity with given its phenomenal accuracy, negligible recoil, and impressive terminal performance. Another non-classic, but at least this one's been around for 20 or so years and has a hardcore fan base. The popularity of 6.5 ManBunn (creedmore) has helped a lot with component availability if you reload.
Of the 3 you list, I'd go with the 243. 357 doesn't have much margin of error for new shooters. 44 can be a bit of a thumper depending on the rifle and load.