IMHO, private tests are good for peace of mind. You don't have to stress out about your upcoming visit to the doctor hoping and praying s/he will allow a particular test and walking out frustrated or devastated if they don't let you get one. My current, stupid, Endo will only do TSH and T4, but that's OK, when I feel things have changed a little bit, or I have increased my dose I pay for a private complete thyroid panel. It's a bit expensive, but you only need a full panel every year. Then you can sit back and listen to the doc's misguided opinions based on your TSH. Of course, it isn't always perfect - like I just ran into - the doc might decrease your dose to a point where you cannot convolute the dose you know you need. Well, you can but you run out of pills in a ridiculously brief time - and your busted. That's when you have to hit the road again and get another doc.
Oh and rT3, yes, I think it is important because it is a double check for you, it lets you know that you might be exceeding your critical, optimal dose. i.e. you are "dumping" T3, i.e. you are getting too much or you are not converting your T4 and you are dumping unused T4. Tricky isn't it!