What is the point you're arguing against? I know perfectly well that conservation of angular momentum is a thing. I know perfectly well that the intrinsic property that we call 'spin' of an electron is really angular momentum. I know that a spinning bucket also has real angular momentum.

So, I'm not sure what you're trying to tell me.

The central point made by colordrops is that angular momentum in a macroscopic object is 100% (accurate to at least ten decimal places) due to synchronicity of linear momenta.

What colordrops wrote at least suggest that he thinks that circular motion and angular momentum are not fundamental but can be expressed or understood in terms of linear motion and linear momentum. As far as I can tell today this is not true, they are independent concepts and one can not be fully understood in terms of the other, neither by looking at circular motion as piecewise linear motion nor by looking at linear motion as circular motion about a point infinitely far away. I only brought up spin because it makes the point pretty clear - or maybe not - that angular momentum is a fundamental concept that can not be recast in other terms, especially it is not just the sum of many linear momenta.

Look at the 2-body gravitational system -- let's say Earth-Moon. Let's put our non-rotating reference frame at the barycenter. At any given moment, Earth has a definite position and velocity, and the Moon has a definite position and linear velocity. If we know these positions and velocities (and masses), we can derive anything else we'd like to know about the system. Including its angular momentum.

This would also be true of any system such that the size is large enough to render the quantum spins statistically close to zero.

Sure but you make an arbitrary choice here, you choose to use Cartesian coordinates. Lagrangian and Hamiltonian mechanics make it more clear that this is more or less an arbitrary choice and that there are other sets of generalized coordinates to describe the problem, something that is in some sense less obvious in Newtonian mechanics because things usually become pretty hard to deal with.

Yes, every choice of coordinates is capable of describing the physics of a system. Yes, you could certainly argue that Cartesian coordinates are a natural choice or in some sense special, derivatives are especially simple and whatnot. But I still think that something is lost when one disregards rotations and angular momentum, they seem to capture an important aspect of the structure of space, its isotropy.

Then again every equivalent description should capture the same things, just maybe not in an obvious way. And then again the existence of spin angular momentum hints at the fact that there is really more to it. As I said, I am really not sure. I used to think of rotations as emergent from translations and I kind of changed my mind but I am probably still on the edge.