then changing the relative airflow on the wings would cause a slip to the left which would in turn change the relative airflow and would help "push the tail around"
It helps to distinguish
turn (changing the direction of travel of the centre of gravity of the aircraft) from
yaw (changing the heading of the aircraft, or as you put it, "pushing the tail around").
Something provides the yawing moment that "pushes the tail around". That can be the response of the fuselage and fin to a slip angle -- which is your "so-called weathercocking effect", or it can be the rudder. Or it can be some of the other effects discussed above.
Those of us who have spent too long flying aircraft with short wings and more power than necessary to stay in the air tend to bank the aircraft and wait for the slip angle to do its job and yaw us around the turn -- hey, we're just mimicking the two-axis autopilot
. Glider pilots, on the other hand, tend to finesse the slip angle to zero with their feet.