But the amount us tiny, current estimates of the dark matter density in the Milky Way result in roughly 700 grams of DM in the Earths volume
I've also seen an article not long ago showing 'visualizations' of dark matter around a spiral galaxy, and it was mostly in teardrop-shaped lobes above and below the galaxy's center and aligned on the axis, rather than distributed through the disk. I didn't think of it at the time but now that I do, it seems like that distribution also indicates that light from the stars is pushing the dark matter away.
If dark mater is a particle, it rarely interacts with anything, including itself; so, it cannot collapse. Most of the particles will be in a constant state of free fall throughout the galaxy. Only if the particle just happens to have a nearly matching velocity would it get trapped in a star or planet. Even then, it would still be wobbling around with nothing to touch.
If so, (I'm no scientist at all), that sounds like a very powerful matter indeed :-)
So yeah, the effect is basically zero.