"...inflammation could be caused by an infection, autoimmune disease, obesity, diabetes or other chronic illness"
Inflammation in the body impairs T3 production in the thyroid and/or reduces thyroid binding proteins.
Although inflammation is only one potential thyroid disrupting event, many more exist like poor organ function, toxicities, deficiencies, stress, lack of sleep, hormonal imbalances, and caloric deprivation.
However, I've seen people that have inflammation for years that gain weight, suffer from fatigue, and other symptoms of thyroid problems. Supplements wouldn't really work for those types of people unless underlying inflammation is first resolved.