They probably won't find Hashimoto's except by testing for the TPO antibodies. There may be a genetic link for Hashimoto's. A number of people on the forum have reported having Hashimoto's as well as celiac disease. The linked wiki article has more info. I thought of it because you mentioned throat irritation after eating gluten. The TPO antibodies test isn't a standard test they give thyroid patients from my understanding. It's something that has to be requested. Hashimoto's is more common in women than men.
Here's another thread discussing Hashimoto's and some other stuff. Has some discussion of thyroid testing in it.
Thank you for the info! I will give it a good thorough read when my kids are in bed tonight