223 and 22-250 will kill deer, but I prefer them more in the hands of someone who knows what they are doing and not a novice.
243 has pretty mild recoil. If you buy a 243 in a heavier gun it will be less.
When it comes to deer, hogs, etc a 223 or 22/250 is (in my humble opinion) an expert's gun. Yes, there have been thousands of game animals killed with the 223, but it requires exacting shot placement. I personally wouldn't saddle a newcomer with that responsibility. Lots of parents with the best of intentions buy a youngster a .410 gauge shotgun without realizing they just handed the child a gun that should be reserved for expert shooters.
The 243 is very capable and the recoil isn't extreme.
And if you reload, it becomes a non-factor.
I haven't used them but there are factory loads for the 243 that are labeled "low recoil". That might be an option.