I can't speak to the low carbing-slower thyroid issue directly. However, I can say that FTSing was my last resort and the ONLY lifestyle that allowed me to
lose weight that I had gained because of hypothyroidism.
I weigh today what I weighed in 2007, shortly before I started having health issues that were eventually traced to my thryoid. I had a thyroidectomy in September of 2008 and continued to gain weight. Last year I switched to a new doctor and he told me to buy "the book." The turkey sandwich I ate before that appointment was the last "carb-ladden" meal I had.
Please don't use thyroid issues as a rationalization. Your thryoid function is fairly easy to monitor with simple blood tests and the benefits of a healthy weight so far out weigh a small fluctuation in thyroid hormone production.
We all need to eat but what we eat is our choice. In my life, FTS is just the way I eat. It's a part of how I live my life.