[Shatner]CONNNNNNNNNNN![/Shatner]
*ahem*
Con is a measure of health, toughness and fitness. One may be physically strong, but may have low stamina, a chronic disease or merely be grossly fat.
Think of it this way- the high Str and Dex PC is the young, raw recruit who is running the army obstacle course...and the high Con PC is the grizzled old drill sergeant who effortlessly beats them through it.
So if you wanted a totally realistic system of stats, you'd probably not only have six or more, but they would have cross over effects (like having a high strength would effectively boost con and dex under certain conditions or something like that). If you are just going for D&D level of realism, I think the 6 stats work fine.
While I think it makes sense to divide health and strength, it is also true that physical strength and conditioning have a very real impact on health. For example, I have asthma. But if I lift weights and do cardio regularly my asthma all but disappears. So there is a direct link between my physical strength and my health (you have to keep in mind stamina is also a function of how easily your muscles tire and how hard it is for them to perform actions). However I also have serious digestive health issues. No amount of weight lifting or running will improve it.
To the OP:
I have a problem with your post here. It has to do with the word I bolded. Cardio is an aerobic endurance workout not a muscular strength workout and is what helps your asthma not the weightlifting. Consider it this way: Doing cardio is like putting points into the Endurance(Con) skill and weightlifting is putting points into the Athletics(Str) skill.
While I think it makes sense to divide health and strength, it is also true that physical strength and conditioning have a very real impact on health. For example, I have asthma. But if I lift weights and do cardio regularly my asthma all but disappears. So there is a direct link between my physical strength and my health (you have to keep in mind stamina is also a function of how easily your muscles tire and how hard it is for them to perform actions). However I also have serious digestive health issues. No amount of weight lifting or running will improve it.
I think I may have been a little unclear in my example. I attribute improvement in lung function to both. That is why I said cardio and weight lifting. For the activities I am involved in I need the weights as well. It isn't just a matter of boosting my lung strength by reducing overall difficulty of doing things in my entire body. I do think cardio is the biggest factor in general for my asthma. But stregthening my muscles (and by that I don't mean simply making them bigger, but improving both explosive power and stamina) through weight routines makes it less strenuous to participate in sports and even to walk up flights of stairs. Which definitely helps the asthma. Quite simply, the easier it is to move my body against resistance the less of a problem it is for breathing. My endurance is definitely bound up in both strength training and cardio. I don't think this is true for all activities. But the sports I am involved in, I definitely see benefit from weights and other resistance training.
What you are describing here is actually weight training not weightlifting.* True weightlifting, picking up the heaviest object you can lift then putting it down, will not improve muscular stamina as much as or as noticeably as weight training.
My only problem with your post was that you implied that direct strength improvement helped handle a constitution aliment when in actuality it was 2 forms of constitution improvement, one of which also had coinciding strength enhancement properties, that was actually doing it.
*Semantics, I know but sometimes they are can be a major influence on how things are interpreted.
I think Constitution is as realistic a stat as any (which is to say, they're all kind of unrealistic*), and it's existence is justified as well as any other. The arguments made in the OP work as well against any other particular stat.
Six stats seems about right to me, so I don't see a need to get rid of any of them.
*In the real world, humans don't have just six things that describe their performance. Real people don't have separate Con and Str, and the things that give one a good (or bad) Con or Str overlap. That's just the nature of trying to simulate a person with only a few numbers that don't interact.

(Dungeons & Dragons)
Rulebook featuring "high magic" options, including a host of new spells.