Crispy, your GP won't test anything beyond TSH. Sometimes you can get them to test T4 but usually they can only do that for people already known to have a thyroid condition. Personally, I wouldn't be happy having your symptoms and not having had a full thyroid panel done.
Sunday...
"Well if you have a low T3 then it stands to reason that you'll have a higher rT3. T4 has to be converted to something."
No, not at all. There are many people all over FB posting thyroid labs with low T3 and mid-range and perfectly acceptable rT3. Having low T3 does not necessarily mean you also have high rT3.
"And if you have a low T3 then there's not much point in paying for a rT3 test as it will be high."
Again, not true. Here are my own labs off meds:
TSH 2.38
free T3 2.6L (3.1-6.

free T4 12.3 (12-22)
Total T4 64L (66-181)
rT3 17 (10-24)
Reverse T3 ratio 0.15
You'll see my free T3 is so low, it is below range. My rt3 is mid-range and would be completely fine if my T3 were decent. Completely disproves what you've just said.
"Also, low T3 and higher rT3 doesn't necessarily mean you have issues with your T4/T3 conversion."
In the absence of any other cause, like fasting, severe illness (being in intensive care), and other obvious causes, the most common cause of low T3 and high rT3 is indeed conversion issues. And when someone has a lot of symptoms of hypothyroidism along with a high rT3, it makes sense to treat the high/low T3 and see if that helps them to feel better.
My original thyroid labs, 3 years ago, were:
TSH 1.37 (0.270-4.2)
free T3 3.9 (3.1-6.

free T4 18.6 (12-22)
rT3 32 (8-31)
And that is indeed what was done for me, I was given a trial of low dose T3 as a result of these labs. If I had not tested rT3 and the only results you were looking at here were the others, I would not have been prescribed thyroid meds with these results. As most functional medicine doctors will tell you, rT3 is an essential test.
Crispy, if you're confused you can have a read of Dr Child's info:
https://www.restartmed.com/reverse-t3/?srsltid=AfmBOorqFGdRS3-OBg4qDHKXlzHateLLJO5TWNXeAn4gQU5gcF0lmMnV