Any type of conditioning is extremely specific, basic SAID. So many scientific studies and practical experience that this point is irrefutable. 100% of all training adaptations for aerobic conditioning happen in the muscle, 0% in internal organs.
General conditioning is a popular training method in many sports and its benefits are obvious when you participate in them. It will improve your recovery, your disposition, it can be used as an alternative method to increase your caloric deficit and give you low-risk opportunity to train in pacing your body. Even if you reject those points, breathing is a skill and alternate activities permit a differing training environment and stimulus. Similarly General Physical Preparedness is often a positive factor towards more specific training.
Try this experiment:
1. run on a treadmill for 3 weeks, notice your improvement.
2. now go run the same distance at the same pace outside. You will be gassed.
I have trained in the gym for outside runs, it is clearly more effective than not training. Being one person I hardly can argue it is as effective as training outside but the training is evident. For instance in a 10k I just ran, I trained in the gym working only on my pacing and breathing but I never went over 2-3k distance. Further I did longer duration HIIT like training that involved intervals of running. This general conditioning was clearly more effective than going into the 10k only doing weight training.
Or try this one:
1. Test Vo2 max on your bike with only your right side of the body, repeat for the left side
2. train only your right side for 6 weeks
3. repeat tests from #1. You will note zero improvement on the left side.
Reverse the 2 and you will notice the exact same thing. There is no such thing as general "conditioning". Your heart and lungs are internal organs. Do you do bike/row/run/whatever for your liver? How about your spleen? Vo2Max is 96% genetic and does not carry over to any other activity.
We are not arguing VO2Max increases, you are the one bringing this up. What is being encouraged here is LISS to reduce someone's breathlessness who clearly already trains lifting.
BodyByScience.net a la Dr Doug McGuff has lots to say on this, very good book.
A sound weights routine such as HST or SS will give you all the conditioning you need. If you want to be good at a particular activity, do that activity exactly in the way you want to be good at.
If you want to limit your training to lifting that is fine. And while I definitely wouldn't consider Rippetoe a major proponent of conditioning I think it should also be noted he doesn't reject it either. From the 3rd edition Starting Strength and starting on page 301 talking about working sets there is a comparison between 1rm exertion and 20rm exertion. Regarding 20rm sets he says, "...the body gets better at responding to the high metabolic demand that is created. Systemic adaptations are primarily cardiovascular in nature, since the main source of stress involves managing blood flow and oxygen supply during and after the set." So while I don't believe anyone was specifically talking about training your oragans, Rippetoe is clearly suggesting you can. Granted Rippetoe has plenty of criticism for his beliefs but to recommend SS as a complete training system that clearly holds a different training suggestions should be noted. While you may feel SS is all that is needed, Rippetoe himself notes that training that isn't SS and is aerobic creates beneficial adaptations.
Further he has a diagram talking about the metabolic systems used in different exercises. Here he illustrates how Rest > Walk > Jog > Run > Sprint progresses from Aerobic to Anaerobic and require different metabolic responses, there he says, "no activity uses only one metabolic pathway, so the illustration above represents a sliding scale of continually increasing intensity of activity." Clearly Mark doesn't argue for much conditioning (and does not include it in his beginner programming) however he clearly does not reject the benefits, as appears to be your stance.
Again, if you don't want to add an aerobic component to you training, great. To argue that adding one is useless however is ignorant.
The reason why "2 a days" in football is 3 weeks long is that coaches have known for decades that 3 weeks is what it takes for the metabolism to adapt to any specific activity. There was an interesting case a few years ago of a man raised in Machu Picchu (high elevation), came to visit the US for 2 weeks, then returned home. He died that night from altitude sickness. This man had metabolic adaptions to high altitude his entire life, and lost it in 2 weeks of changing his environment.
So it takes 3 weeks, unless it takes 2 weeks and you die? Kidding. I am going to assume you are trying to argue something very reasonable and I simply don't get it.