Thanks for the questions and I completely understand your confusion.
The first thing to understand is that HR is a very imperfect measure of intensity. Its the best one for mountain runners, skier and alpinists/mountaineers. But, for rock climbing HR does not reflect the work load well at all unless you are climbing lower angle easy to moderate terrain where you can move fast and are climbing mostly with your legs. On steep and difficult (for you) climbs HR is nearly useless in terms of monitoring the training effect. This is because a healthy/fit heart has the capacity to supply a lot of blood for locomotion. The leg/hip muscles are much bigger than the arm/shoulder muscles so demand for blood/oxygen to the legs is much greater than the arms. So, the cardiac response will not be similar to running when you are climbing.
Perhaps I am wrong in my interpretation of you question. You state that you are like the sedentary person on the treadmill. Do you mean you are literally gasping for air? Or do you mean that you get a serious pump in the forearms? I ask because……
A healthy and reasonably fit person should not be limited by the heart’s capacity to supply oxygen to the upper body climbing muscles on hard rock climbs. Instead that person will be limited by those muscles’ ability to utilize that oxygen in the most aerobic way possible. And for most climbers of steep routes the limitation will come down to the grip strength and endurance of the forearm muscles. These are pretty small muscles and from the above discussion it should be obvious that the heart has no problem providing enough oxygen to them. This makes the LOCAL MUSCULAR ENDURANCE of the forearms one of the most key factors in determining your climbing success.
Developing local muscular endurance is something we target specifically in some articles on this site. Those articles are directed at developing LME in the big locomotion muscles needed for mountains. But similar principles exist for building LME in the forearm/shoulder complex.
30 minutes of low intensity running is going to have minimal aerobic training benefit for most healthy and reasonably fit people because the locomotive muscles are already aerobically fit from spending years moving you around. Remember that the fitter you become the harder it is to become fitter. It takes more and more training stimulus to make gains. As an example: What for you would be an easy bouldering workout, that didm’t give you almost any training stimulus would completely destroy me I am sure.
Your forearms especially do not have that aerobic training history of your legs. So, they do not have that same aerobic base. It will take much less stimulus to give them a strong training effect. That’s why 30 minutes of ARCing is going to be a powerful stimulus towards aerobic development in the forearm muscles (which are the main limiters to your ability to hang on).
Running, while it definitely will be good for your general fitness and overall health will do almost nothing to improve your hard rock climbing specific endurance.
Rock climbing specific training, be it strength or LME, does not lend itself to using Training Peaks hrTSS (heart rate Training Stress Score). This is true of full body strength training but doubly true when it comes to forearm training. The Anderson brother’s book The Rock Climber’s Training Manual does a very good job of explaining proper rock climbing training methodologies and theories. It also shows good ways to monitor progression in training.
You no doubt have ample strength and just need to allow adequate time to build an endurance base.