Welcome to the Thyroid forum, GroverCat.
Thyroid hormones serve the purpose of regulating the body's metabolism. They can affect heart rate, respiration, sweating, hair growth and many other functions including digestion. Certainly they can increase or impede motility and this can affect reflux syndromes.
I had Graves rather than Hashi's, so I was wildly hyperthyroid. In addition to doubling my normal heart rate and many other symptoms, it caused dumping syndrome due to increased motility. I would have episodes of severe diarrhea lasting about three hours and followed by a single episode of forceful vomiting. Later, it started causing vagal symptoms - I would get faint, drenched in sweat and recover only to be met an hour or so later by the GI symptoms.
Yes, our thyroids can affect almost any body system. Glad you found your problem.