I mean that’s all well and good until you have an object in your game and you’re like “I’d like this object to be leaning at 45 degrees, oops I mean 0.70710678118 + i*0.70710678118
If you use angle measures (of whatever units), when you say "rotate by an eighth of a turn" you are instead going to end up with something internally like: multiply some vector by the matrix
[cos ¼π, –sin ¼π ; sin ¼π, cos ¼π]
which is ultimately the same arithmetic, except you had to compute more intermediate transcendental functions to get there.
If you just have to do this a few times, angle measures (in degrees or whatever) are a convenient human interface because most people are very familiar with it from high school. You can have your code ingest the occasional angle measure and turn it into a vector-relevant internal representation immediately.
P.S. If you write 1/√2 in your code compilers are smart enough to turn that into a floating point number at compile time. :-)
(Game engine developers are smart people and have lots of practical experience with the benefits of avoiding angle measures, as do developers of computer vision, computer graphics, robotics, physical simulations, aerospace flight control, GIS, etc. etc. tools.)