Enough – you are correct on the T3 and T4. Thyroid hormone binds to protein in our blood and is "bound". The thyroid hormone that is not bound to protein is "free" and is the most active in our systems. The answer to your question on why your doc measures T4 may rest with your lab. Some labs report a total T4 (free and bound). T4 is used sometimes to monitor replacement.
I just posted this on another thread – I just found out my TSH is normal but I feel hyperthyroid. My PCP checked a T4 and it is elevated. TSH, however, it the usual way to monitor replacement (not T 4) because TSH is more sensitive.