The question came up when I was searching for the reason why earth moves faster when it gets closer to the sun.

The Earth moving closer to the sun can be explained using conservation of angular momentum. Because gravity is a central force, there is no torque and therefore angular momentum is conserved:

$L = mrv$
$v \propto \frac{1}{r} $

But gravity follows inverse square law:

$\frac{GMm}{r^2} = \frac{mv^2}{r}$
$v \propto \frac{1}{\sqrt{r}}$

So rate of change in velocity in order to maintain rotational motion is not consistent with rate of change in velocity due to conservation of angular momentum, what am I missing here?

Additionally, what is the connection between the two? So does gravity cause increase in velocity? Does conservation of angular momentum? Is it both? Are they related? I'm sorry if the question is stupid but I am very confused.