I think the idea of covering the thyroid gland from the sun came from the topic of thyroid cancer. Like it happens from radiation. But the radiation from the sun is not the same as thyroid cancer.
Problems would be from ionizing radiation with some kind of radiation therapy for oncology. This is when something is deliberately irradiated on the head and neck, and the thyroid gland is hit.
Or if you ingest radioactive iodine after an accident at a nuclear power plant. But not from the sun. The sun will not reach the thyroid gland.
I have a suspicion that someone after thyroid surgery was forbidden to expose a sore spot in the sun. But this was done not because they were afraid of harm to the remains of the thyroid gland, but probably to prevent skin pigmentation in the area of the postoperative scar.
The fact is that any scars and even traces of ordinary scratches need several months, or maybe a year, to be protected from the sun. Otherwise, the scar may become pigmented.
Remember history about adrenal insufficiency? There, if fresh scars were pigmented, then they remained so forever, because the pigment is not washed out from the scar tissue. Here's a story ...
I also have a suspicion that vitamin D or some kind of hypersensitivity to light in diseases of the thyroid gland may have been involved in this bike. Not sure. Have you heard similar explanations?