You are comparing two different situations.

When the horizontal force $F$ is applied to the bob it stretches the string so that there is tension $T=\frac{mg}{\cos \theta}$ in the string, and thus the vertical component of $T$ is equal and opposite to the weight of the bob. By equating the horizontal component of $T$ with $F$ you can find the angle $\theta$. When the bob is released from this position it does indeed initially accelerate horizontally because the horizontal component of $T$ is now not balanced by $F$. As the bob moves horizontally it reduces the tension in the string.

But if the the bob is swinging freely, then at the extreme point of each swing we have $T=0$ because the angular speed of the bob is instantaneously zero. The only force on the bob at this point is its weight, so at this instant the bob accelerates vertically downwards (the diagram has got the angle of acceleration wrong). As the bob moves downwards slightly, it stretches the string, which creates tension in the string. This tension provides the centripetal force that now keeps the bob moving in a circle.

You are comparing two different situations.

When the horizontal force $F$ is applied to the bob it stretches the string so that there is tension $T=\frac{mg}{\cos \theta}$ in the string, and thus the vertical component of $T$ is equal and opposite to the weight of the bob. By equating the horizontal component of $T$ with $F$ you can find the angle $\theta$. When the bob is released from this position it does indeed initially accelerate horizontally because the horizontal component of $T$ is now not balanced by $F$. As the bob moves horizontally it reduces the tension in the string.

But if the the bob is swinging freely, then at the extreme point of each swing we have $T=mg \cos \theta$ because the bob is accelerating tangentially, so the net radial force on it is zero.