In the equation:
$$ \frac{\partial \mathcal{L}}{\partial r} - \frac{d}{dt} \frac{\partial \mathcal{L}}{\partial \dot{r}} = 0 $$
It seems to rely on $ r $ and $ \dot{r} $ being independent but we know that one is a derivative of the other so how can we state that they are independent. I have seen it treated this way in the example of $ \mathcal{L} = T - V $ where $ T = \frac{1}{2} m \dot{x}^2 $ and $ V = mgx $. A lot of videos state that taking the derivative $ \frac{ \mathcal{L}}{dr} $ of a term that contains only $ \dot{r} $ (not r) means we can assume that that term goes to zero but how do we know this is true?
