Due to having Hashi's (I was told belatedly), once I started on Synthroid, it was continually increased until I was taking a full daily replacement amount of 200 mcg. With that, my thyroid gland was basically inactive. That has been well over 25 years now, with no noticeable effect on my thyroid gland, that I know of.
As you start to take thyroid meds, TSH and your thyroid gland's output of natural thyroid hormone, is reduced. The net effect of this is often no change in total thyroid hormone in your blood until the TSH is suppressed enough that natural thyroid hormone production is stopped. At that point further increases in thyroid meds will raise your total thyroid hormone levels and start to provide symptom relief. That is what happened with me. For many it is an unavoidable effect of taking enough med to relieve symptoms.
I don't know if the doctor's statement that dead organs turn into cancer has any basis in fact, or if it is even a remote possibility. I haven't yet found any info that confirms that it should even be worrisome.
I certainly don't agree that when an organ "dies" or no longer function, that it turns to cancer. Many of us have Hashimoto's and our thyroid is no longer working, but we don't have cancer.
When women go through menopause, their ovaries no longer work and not all of us get cancer. Other organs can stop working, too and people don't get cancer from it.
Very rarely, is a thyroid removed, simply because it no longer produces hormones and the only thing you can do when the thyroid no longer functions properly is to replace the hormones it produced with medication, whether it be synthetic or NDT.
Even if you got your thyroid out, you will have to replace the hormones with medication, because you can't live without them.