This is at best questionable and at worst false.
If you only want to use sin and cos as functions for doing trigonometry, it is true that you can choose whatever angle unit you like and stick with it and it will be fine.
For most other stuff, e.g. differential equations, complex analysis, signal processing and mechanics, it's pretty much inescapable that the zeroes of sin are at integer multiples of pi, and that's that.
It provides an easy way to connect the complex exponential with trigonometric functions (and everything you get from that, i.e. Taylor series, nice behavior in diffeqs). You can do the same in degrees as well, but you end up with weird conversion factors with pi in the denominator, a strong indication you should have multiplied by pi to begin with.
sin'(ax) = a cos(ax)d/dx sin(2 pi x) = 2 pi cos(2 pi x)
vs
d/dy new_sin(y) = 2 pi new_cos(y)
“Fundamental unit” really depends on what you care about.
It doesn't help that at school our first look at sin and cos is all about adjacent sides and opposite sides in right-angled triangles. It's understandable, because jumping straight into the deep end would be too hard, but it's a bit misleading.
In most mathematical applications, the x in "sin(x)" doesn't even represent an angle, so it doesn't make sense to talk about whether sin and cos are "in degrees or in radians". They're simply functions that crop up as solutions to the differential equation that describes harmonic oscillation; or the imaginary and real parts of e^ix; or exponentiation of certain matrices; or a whole load of other stuff I haven't thought of.
In all those settings, it turns out that sin has zeroes at integer multiples of pi, which forces the convention that a half-turn is an angle of pi, and the definition of radians follows from there. But as I said, for the specific case of basic trig, carrying around a scaling factor and doing everything in degrees is easy enough. Carrying that same scaling factor around in pretty much any other application of sin, cos and related functions would be hell.
For that particular application, x is exactly the argument[1] of your complex number, though!
[1] https://en.wikipedia.org/wiki/Argument_(complex_analysis)
Interpretation is often strongly motivated by what "comes first" in the order in which you learn things, so the whole debate is a bit subjective anyway. Another common example: at school we learn the integral is the area under the curve. At university we learn the area under the curve is the integral. The integral is the "real" thing and the area is just a convenient geometric interpretation, which actually makes no sense for many (most?) integrals. At school we learn it "backwards" purely because it's easier that way, and visual aids are helpful. I think something similar applies to sin and cos.
Could you elaborate on this? Because we defined the (Lebesgue) integral in my analysis 3 course exactly in this way: First define what measurable sets are, and what their volume is. Then the integral of a non-negative function is the volume under its graph, if that is measurable.
I studied engineering, and pretty much everywhere where we needed the radian form with Pi, the mental reasoning was "one cycle, or repetition, or loop ot whatever is Pi". Never did Pi have any deeper meaning that helped understand the logic of the problem.
It’s also true that they are the basic building blocks of cyclicity.
But to say they are not geometric tools is dishonest. They instead show us that geometry is deeply connected to many other, sometimes-surprising, areas of mathematics.