Let imagine that we put wind indicator on rotating merry-go-around.
Will wind indicator stay perpendicular to the radius of circle,showing
tangential airflow velocity?
Assuming that the indicator has a mass distribution such that the apparent "centrifugal force" from the rotation has no effect on it, and also assuming that the physical length of the wind indicator is trivially small compared to the radius of the circle, then yes, the wind indicator will remain perpendicular to the radius of the circle at the point where it is located, showing that the "relative wind" is tangent to the circle.
A very lightweight piece of yarn would do essentially the same thing, though there would be a very slight tendency for the apparent "centrifugal force" to displace the free end of the yarn toward the outside of the circle.
However the problem gets more tricky if the length of the wind indicator is not trivially small compared to the radius of the circle. If the length of the wind indicator is not trivial, this means that the "vane" or fin is located well aft of the weathervane's pivot point. The relative wind at every point along the circumference of the circle is aligned with the line that is tangent to the circle at that point. So when the vane streamlines itself to to the relative wind which is tangent to the circle at a point well behind the pivot point of the weathervane, this means that the weathervane is not parallel to a line drawn tangent to the circle at the pivot point of the weathervane. Rather, the weathervane is skewed with its nose outboard, and its tail inboard, of the line drawn tangent to the circle at the pivot point of the weathervane.
Your intuition is correct that this is why tend to see some sideslip in circling flight, especially in slow-flying aircraft whose linear dimensions are not trivial compared to the radius of turn. The vertical fin tends to streamline itself to the relative wind, which means that the fuselage tends to be parallel to a line drawn tangent to the circular flight path at a point well aft of the CG.
This effect is illustrated in section 8.10 of the "See How It Flies" website.
In short, in turning flight, the "relative wind" is curved, rather than linear. It curves to follow the curvature of the flight path.