Net force when deriving relation between torque and angular acceleration

When deriving the equation $\tau = I\alpha$, it is assumed that $$F_{tangential}=ma_{tangential}$$$$a_{tangential}=\alpha r$$$$\tau=rF_{tangential}$$$$\tau=mr^2\alpha=I\alpha$$

However, doesn’t Newton’s second law describe that $\Sigma F=ma$? And isn’t $\Sigma F$ in this case equal to $F_{centripetal}+F_{tangential}$? Why don’t you have to take into account of centripetal force?

Or you can just arbitrarily throw around $F=ma$ for any force without minding that it’s not the net force?

To put it in coordinates, let's choose a $(r,\theta)$ coordinate system. In that system $$\vec{F}=F_r\hat{r}+F_{\theta}\hat{\theta}.$$
To calculate the torque about the origin that this force exerts on a particle located at position $\vec{r}$ we calculate a cross product $$\vec{\tau}=\vec{r}\times\vec{F}.$$
In case you don't know about cross products, one of the results is that any component of $\vec{F}$ that is parallel to $\vec{r}$ vanishes, and because $\vec{r}$ points in the $\hat{r}$ direction, the $F_r$ component doesn't contribute to the torque.$$\vec{\tau}=rF_{\theta}\hat{k}$$