If $F_f=mg\sin\theta$ then there would be no net force down the slope and hence no linear acceleration of the ball down the slope.
Note that that the FBD is incorrectly drawn.
As the ball rolls down the slope without slipping the centre of mass of the ball undergoes a linear acceleration and there is also an angular acceleration of the ball.
As drawn there is no torque about the centre of mass of the ball and so there can be no angular acceleration of the ball.
The point of application of the frictional force $f$ must be moved as shown below.
In this case it is fairly obvious as to the direction of the frictional force but it is worth a little consideration as for some problems that direction is not quite as obvious eg a ball rolling up a slope.
If the ball slip down without rolling its acceleration would be greater than if the ball was rolling with no slipping.
In terms of energy the ball now converts its loss of gravitational potential energy into both linear and rotational kinetic energy so its final linear speed would be less when rolling.
That means when rolling the net force down the slope acting at the centre of mass must be smaller than when there is no rolling.
Thus the frictional force must act the slope.
The angular acceleration of the ball is in the clockwise direction hence the torque about the centre of mass must be clockwise again indicating that the frictional force is up the slope.
One can now set up two equations $F=ma$ and $\tau = I \alpha$ and with the no slipping condition $a=r \alpha$ solve a problem.
It is true that if the slope is too steep and/or the coefficient of static friction is too small the ball will roll and slip under the action of a kinetic frictional force ie the no slipping condition cannot be satisfied but usually this is not the case.
Perhaps you misheard your teacher?