I have been on a similar quest and have turned over in my mind all the various setups a hundred times. My concept differs slightly in that I consider mine a "homestead gun" of sorts and meant to be cheap to feed, effective, versatile and easy to acquire components for and to cast and reload for. There are a lot of cartridges that would do well in terms of performance, but logistical considerations have to come into play as well as versatility in terms of utility loads. I've done this dance with the .357 Max, 7.62x39, .223, 30-30, .44 Mag,.... These are all compromises one one end of the spectrum or the other - or both, but for ONE rifle, there will be compromises which must consider each individual's perceived condition.
For where I am and what I need to do, I consider the .357 Mag to be the best combination of compromises.
Even though I have never been a huge fan of the .357 mag as a handgun round, there is little to dispute in its performance, popularity and portability. I cannot deny its due as a big player in the handgun.
When I started using it in a carbine, it got really interesting.
Bear in mind that these points are based on my own personal experience and where I live, what I think I need to do - my situation, my condition. They may not be the same for everyone and I am not listing all the details of all conditions under which each point makes sense to me, and, yes, other cartridges share many of these points. These are all relative statements and I am not arguing against any other choice.
Brass has been easier to find and cheaper than other cartridegs I have considered;
I have a big bag of .38 Spl. brass form when no one wanted it;
Brass seems to last a long time without trimming or annealing;
Mixed headstamps have never caused me any problems at all;
Ballistically, it should be "enough" for deer if that situation/opportunity were to arise;
It is about the easiest thing to cast for and I have cast for a lot of stuff much bigger and much smaller;
Gas checks are not even necessary even with top loads (but I can make them if I have to);
It does not require large amounts of lead for wide range of useful weights of bullets;
A very wide variety of bullet mould weights and shapes are available and I have several already, mostly LEE, which are inexpensive;
Tumble lube is effective even at the higher velocities of which it is capable;
Sizing bullets, especially the TL type is seldom required - seems more forgiving than smaller calibers in this respect;
It is not particularly fussy about the alloy or hardness I use;
Shoots even imperfect bullets quite well;
Reloading is quick, easy and requires no case lube with carbide dies - if you need to size the case at all, like with light loads;
It is actually difficult to come up with a poor load - accuracy is usually at least very good and can be excellent;
The variety of powders suited to it is very large and one is bound to have something that will work in it;
Does not use a lot of any of these powders;
I can use small pistol or small rifle primers, and I use only small rifle primers anyway;
A somewhat lower end of optimum diameter bullet for non-expanding projectiles;
I can get a bunch more of these in my pocket than most other rounds I have considered;
Recoil is negligible and I am a consummate recoil whimp (neck injuries);
It's cheap enough/easy enough on the lead stash that I can still "play"/practice a little and it IS fun to play with;
It comes in some very lightweight, portable and handy carbines;
Anyone in my household can shoot it well and with confidence;
Is suitable to regular sights for the distances I feel comfortable shooting it;
Very much at home with low-powered scopes or red-dots;
I can shoot it off-hand at up to a hundred yards and hit a rabbit-sized target;
I could probably shoot it 25 or 50 yards beyond that with a rest and practice.
It may not be the best for elk and it may not be practical for a four-hundred yard coyote shot, but one could lay down some very disconcerting cover fire with it if pressed to do so. The 30-30 is probably a "better" round for dedicated deer hunting and the 223 is certainly much better for a dedicated varmint round, but I consider the .357 Mag to be the best combination of compromises.
I dearly love my custom Mausers in bottle-necked, long-range chamberings, but I have to give the nod to the .357 mag. for this usage category.