Any physical quantity, for instance length, can appear as an argument of a nonlinear function that can be developed in a Taylor series. So your example would be identical for any other quantity not only for angle. I can make an analog computing element where a voltage is equal to the sinus of another voltage, so after your theory, voltage is dimensionless.
The reason why this is possible is that the arguments of such nonlinear functions are either explicitly or implicitly not the physical quantities, but their numeric values, i.e. the ratios between those quantities and their units, which are dimensionless.
In the case of the nonlinear sinus function, what is usually written as sin(x) is just one member of a family of functions where the arguments are angles implicitly divided by units of plane angle:
sin(x) is the sinus function with the angle implicitly divided by 1 radian
sin(x * Pi/2) is the sinus function with the angle implicitly divided by 1 right angle
sin(x * Pi*2) is the sinus function with the angle implicitly divided by 1 cycle a.k.a. turn
sin(x * Pi/180) is the sinus function with the angle implicitly divided by 1 sexagesimal degree
It is very sad that the logical thinking about angles of most people has been perverted by what they have been taught in school, which is just a bunch of nonsense copied again and again from one textbook to another.
This to me sounds like the most natural explanation. For example, in a sibling comment someone mentioned that "you can calculate e^(-t)", but I disagree: in physics it's always e^(-t / T), where T is some time constant, so that the argument of the exponential is dimensionless. Same applies to sin(x): usually we write something like sin(2pi f t), where the units of f and t cancel out, and the 2pi is there to cancel out the invisible implicit 1 radian. sin(ft) would be wrong, at t = 1 / f you wouldn't have advanced by a full cycle.
360 - (360^3)/6 = -7M degrees
2*pi - (2 * pi)^3 / 6 = -35 radians = -2k degrees
1 - (1^3)/6 = 0.8 turns = 300 degrees
As far as you three examples go, which is "correct" depends on what you are trying to calculate - if you want this to approximate the power series for sin close to 0 you should use radians. Otherwise you use something else.
sin(x)
= x/(1 radian)! - x³/(3 radians)! + …
= x/(1 radian) - x³/(1 radian × 2 radians × 3 radians) + …
There is a natural reason for pi occurring in physics that makes little sense to ignore.
Treating it as something to be dealt with misses the forest for the trees.
Radians are the naturally occurring Euclidean unit of angular measurement.