Metabolism is definitely key to improving health. A restrictive diet that makes you cold and slows digestion will begin to cause more allergies and bacterial overgrowth in the small intestine. This isn’t a guaranteed effect of the protocol though. It depends on where you’re starting from metabolically, and how you end up implementing it. This is why it’s important to avoid jumping into anything. Some people seem to do well, and it seems to be the people eating the most foods and getting a good amount of calories.
What kind of exercise do you do? Endurance exercise can lower your pulse, but I don’t think anaerobic stuff (sprints, weights) would. If you’re running for long periods, you may have identified one source of your problems. Endurance exercise is literally stress. Just check your pulse 30-minutes after a workout and you’ll see stress hormones are still driving it high, assuming you’re still able to produce them adequately. Active thyroid hormone reaches zero after about 40-minutes of running.
Part of the problem with doctors is that they specialise in pathology. Like you mentioned, a TSH of 4 is considered the “limit” for hypothryoidism, but health doesn’t work on these arbitrary thresholds. If you have a disease, you’ll be able to get some form of treatment, but if you want your failing “within range” health optimised, you’re probably going to have to start researching and experimenting, using your doc’s help as much as you can. What are other reasons your pulse and temps are low, or what other explanations are there for hair loss and depression? Are there any biomarkers you can look for in tests? What links all these together? Can you influence them with diet, lifestyle changes, medication or supplements?