You're exactly right about doctors missing thyroid disease by relying too heavily on certain tests. It's very important to pay close attention to the symptoms a patient is presenting. Also, an often-overlooked test that's a very good indicator for Hashimoto's disease (one type of hypothyroidism) is elevated TPO antibodies. See my article on how your doctor might miss your hypothyroidism: https://www.empowher.com/news/herarticle/2009/09/17/are-you-antidepressants-when-real-problem-your-thyroid-gland?page=0,0
I also agree that stopthethyroidmadness.com has great information. It's a site most doctors could profit from, as well, to gain more information on hypothyroidism.