I would assume that if there is something physically wrong with your thyroid, flushing wouldn't help it. But if all your hormones are out of whack, and this has affected thyroid production, then I think flushing could help a lot. I don't have any scientific info to confirm this , this is just my gut feeling. I feel flushing has indirectly help me with my adrenal problems, even though it doesn't affect my adrenals directly.
But regardless of whether it would actually improve your thyroid directly, it would probably help with your recovery overall. It might be one step of many, but it is an important step.