So, as I have understood, the unit circle lets mathematicians define the functions sine cosine and tangent on a way that allows them to have the value of any real input, which is interpreted as the angle created by traveling along the circle counterclockwise with the X axis.

This is wrong. In Calculus (aka Real Analysis), the domain of the sine and cosine functions is the Real number associated with a specific arc length. Since the circumference of the unit circle happens to be $(2\pi)$, and since (in Analytical Geometry or Trigonometry) this translates to $(360^\circ)$, students new to Calculus are taught about radians, which is a very confusing and ambiguous term.

Such students are taught that $(2\pi)$ radians equals $(360^\circ)$, and so the (new) Calculus student continues to regard the domain of the sine and cosine functions as angles, measured in radians.

Then, as you progress in Calculus, you start encountering Math problems that render the above paragraph nonsensical. One example is the Taylor series for the sine and cosine functions. Another (classic) example is

$$\int_0^1 \frac{dt}{1 + t^2} = \text{arctan}(1) - \text{arctan}(0). \tag1 $$

At this point the student becomes confused, because the LHS of (1) must evaluate to a Real number, while the student's initial understanding of radians causes the RHS to evaluate as the difference of two angles.

Then, the student realizes that they have to abandon the notion that a radian represents the measurement of an angle. Instead, the student learns to regard the radian as a Real number. That is $(2\pi)$ radians equals the circumference of the unit circle.

With this understanding, the student realizes that the RHS of (1) above evaluates to $(\pi/4) - 0 = \pi/4,~$ which is a Real number.