Why does a clean configuration lead to the generation of greater wingtip vortices?
I would actually assume the opposite to be true. 

Let us assume the vortex intensity and the induced drag are directly correlated. The induced drag coefficient can be expressed as:

$c_{Di}=\frac{c_L^2}{\pi\cdot AR \cdot e}$


$c_L$: Lift coefficient
$AR$: Wing aspect ratio
$e$: The so-called Oswald factor, a number which equals one for elliptic lift distributions. It decreases for distributions that deviate from the elliptic one.


Most wing designs aim to achieve an elliptic-like lift distribution in order to minimise induced drag in clean configuration. 

As flaps, spoilers, air-breaks and so on are deployed the lift distribution will deviate substantially from the elliptic one (see diagram), hence reducing the $e$ factor and increasing the induced drag. If the first assumption still holds, an increase in induced drag will have been caused by an increase in vortex intensity.



Additionally, significantly higher lift coefficients can be achieved with deployed high-lift devices, which also contribute to increase vortex intensity.