The idea of the radian isn't a particularly hard concept.
Sure, pi is an irrational number. But if you know what pi is, which primary schoolchildren are supposed to have covered when they learnt how to calculate the area and circumference of a circle, learning radians shouldn't be tricky either. Yet students don't cover radians until well into secondary school (C2 to be precise). Why is this?
Plus I've heard that radians have significant advantages compared to degrees in mathematics and mathematics related fields.
Yes, I know. Only on TSR would you get these sorts of arguments.
I'm anticipating accusations of my trying to indoctrinate schoolchildren into STEM subjects.
Degrees are more practical for most of the population as most will only use them for measuring angles where the 90 units to a right angle makes a lot more sense to use than the 1.57 units to a right angle that you get with radians. Radians only really become more useful when looking at ratios.
In Scotland the National Qualifications, equivent(ish) to GCSEs, feature two strands of maths - Maths and Lifeskills Maths.
The first is much as usual, with pythagoras and trigonometry etc, whilst the second bypasses that stuff and focuses on (dramatic pause) "life skills".
This is a great option to be have, although of course the problem is at what stage is it appropriate to split pupils down either path.
Relatively speaking, the ability to express the most commonly used angles in easily expressed numbers, as opposed to a system that involves using a greek letter to represent a non-natural number to express the same fractions. How far is a quarter turn again? Oh yes, half-Pi, of course.... How do I know if this angle is more or less than perpendicular? Well, just look at the scale and compare your answer to Pi/2...