[Epistemic status: speculative. Definitely don't try to make a decision based on this without speaking to an endocrinologist first.]

So, let me see if I understand what you wrote, adding in a few things I read on Wikipedia and the interpretations that seem obvious to me.

T3 controls metabolic rate, by upregulating metabolic processes throughout the body. TSH controls the concentration of T3 by setting the rate at which T4 is converted to T3. TSH is tested for, T3 and T4 are usually not. The Wikipedia page for TSH lists diagnoses for the cross-product of T3 and TSH, with primary hyper- and hypothyroidism corresponding to the cases where they are mismatched: high TSH and low T3, or low TSH and high T3. Cases where T3 and TSH are both low indicate iodine deficiency, because iodine is also a necessary part of the conversion from T4 to T3. TSH is linked to the circadian rhythm.

Adding a bit of interpretation of my own, TSH represents the difference between the body's overall metabolic rate is, and what some mechanism thinks it should be. Under this model, symptoms of metabolic-rate-too-low would appear if:

• That unspecified mechanism were disrupted such that TSH was targeting a level of activity that was too low
• There is an unaccounted energy sink which the thyroid system can't detect/compensate for
• The circadian rhythm was disrupted in such a way that TSH was inconsistent from day to day

(All diabetics with imperfect blood sugar control would fall in the "unaccounted energy sink" category. I have T1DM. fibromyalgics probably would too; the characteristic symptom of fibromyalgia is chronic pain of undiagnosed origin, and chronic pain is very likely to have a corresponding ongoing energy expenditure.)

At this point the selection of possible causes has fanned out enough that it seems implausible for everyone with CFS symptoms to have the same root cause. But it's also the case that, under this model, T3 supplementation is likely to help with a broader range of causes than TSH is.

However, there are two good reasons to hesitate before trying a T3 supplement such as pig thyroid. First: this is bypassing several feedback/regulatory steps in the body, so there's a much higher risk of accidentally overshooting and getting a dangerous overdose. And, second: increasing overall availability of energy in the body can make infections and cancers worse.