Would a thyroid groider push on my vocal chords and permanently damage them?

Yes: A goiter is an enlarged thyroid gland sits beneath the trachea where vocal cords (vc) reside. The nerve that tells the vocal cords to function is the recurrent laryngeal nerve. It's course travels from brain around thyroid then up to vc. A lrge goiter or nodule can stretch the nerve causing damage, why its recommended to remove nodules by 4 cm can any voice change taken seriously.