The short answer is that it pulls the Sun and the planets by the same amount, so their relative motions within the solar system can ignore what's going on outside.

The dark matter halo of the Milky Way is very big and smooth. It's a big roughly spherical blob, that's denser in the middle, and less dense on the outside. Its net effect is that it an adds extra gravitational force towards the centre of the galaxy.

But this is all just classical gravity. The weirdness is that dark matter doesn't interact electromagnetically (although neutrinos don't do that either), but it's totally normal in how it interacts with gravity. If you had a big blob of gas with the same mass and distribution as the dark matter halo, it would produce exactly the same gravitational force (although this gas halo would be unstable, and would cool and collapse, as it can interact electromagnetically). The reason dark matter makes stars and gas orbit faster than you'd otherwise expect is just because there's more mass, and more force, and this mass is more spread out than the stars (which are more heavily concentrated in the centre), which means that the force drops off less slowly with distance than it would otherwise.

So the answer for why dark matter doesn't affect the orbits of the planets in the solar system is the same as why the combined gravity of all other stars and gas and black holes and everything else in the galaxy doesn't affect the orbits of these planets, or even why the Sun doesn't affect the trajectory of a football on Earth. If the gravitational force field varies only on a much bigger scale than the system you are looking at, then it pulls all objects in that system by the same amount. This means that, from within the system, you can't really tell that you're getting pulled at all, other than by looking at the overall motion of the whole system.

There is a middle ground where you get "tidal forces". This is where gravity differs a bit across some object. So, the Moon does pull the near side of the Earth more than the far side, which pulls Earth's oceans a little bit. But this is much smaller than the net total effect of the Moon on the Earth as a whole. But the gravitational force of the galaxy is on a way bigger scale than that, and we really can just ignore it if we only care about what's going on inside the solar system.

The smoothness of the dark matter isn't the full explanation. A completely uniform distribution of dark matter would affect planetary orbits if there were enough of it. It would source a harmonic oscillator potential.

The main reason dark matter does not significantly affect planetary orbits is that there is simply not enough of it. Stars and star systems form when gas cools and condenses. Since dark matter does not cool, it does not participate in this process. Consequently, whereas the average density of the solar system within Neptune's orbital radius is about 3*10^12 GeV/cm^3, the average density of dark matter in our neighborhood is about 0.4 GeV/cm^(3), ten trillion times smaller.

In the early universe, both the gas and the dark matter were cold. With little thermal motion, both components were able to participate in the gravitational clustering that formed galaxies. Consequently, there are comparable amounts of dark matter and ordinary matter in galaxies. That means dark matter contributes significantly to gravitational dynamics.

When material falls into galaxies, it gains kinetic energy, becoming too hot to participate in further gravitational clustering. Gas can cool, e.g. through inelastic collisions, but dark matter cannot. Star systems, which form inside galaxies, thus form in an environment where the dark matter is hot but the gas is cold. That means the gas can participate in the gravitational clustering but the dark matter cannot. Consequently, star systems have much more ordinary matter than dark matter, so the contribution of dark matter to gravitational dynamics is negligible.

The simple fact that dark matter seems to have coalesced in blobs means it must feel some interaction besides gravity, right? Otherwise it would not be able to 'cool down' and collect in a single place? Or could normal matter do a sort of 'sympathetic cooling' for dark matter?

Not necessarily. Dark matter cools in the early universe due to cosmic expansion (it's the same phenomenon as cosmological redshift, but it's even more efficient for nonrelativistic particles). The fact that dark matter clusters in galaxies does give us information about what dark matter could be, though! For example, it can't be primarily composed of ordinary neutrinos (which are technically dark matter), because they would be too hot to reproduce the observed clustering.

So yes, that's another good point. Part of the answer is a gravitational force can affect a system without affecting the motions within that system - and that does come down to the smoothness of the potential. I also wanted to clarify that dark matter isn't a magic thing that just speeds up orbits - in terms of gravity, it's just a distribution of matter, that acts like any other distribution matter. But yes, the other part is that there isn't an additional local effect, as the local density of dark matter really is very low.

Though I would want to clarify for other readers (as I assume you know this of course) that while there is locally not much dark matter (it has a very low density), but because it's smoothly spread out over such a huge 3D structure, its total mass is much larger than the mass of the stars and gas in the Galaxy.

I'd like to try to simplify. The scale of the solar system compared to the galaxy is just so miniscule that the dark matter within the solar system is miniscule compared to the the overall amount throughout the galaxy. The existence of Dark matter was discovered by the difference in the way things orbit a star compared to how stars orbit a galactic center. Within a solar system the farther from the center you get the slower the orbit of an object becomes. Whereas the galaxy of as a whole rotates ad if it were a plate, the farther out you get the faster the tangential velocity gets so the orbital period remains roughly the same, this defies conventional physics and can only be explained by adding additional unseen mass impacting orbits the further out you go from the center.

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Thank you for this extensive explanation but unfortunately calculation of gravitational interaction in symmetrical smooth objects is I think independent of configuration.

All matter outside of an object's orbit should affect it equally, while that inside can just as well be treated as a point object at the barycenter.

A better answer to the original question is to say that the fraction of dark matter to visible matter within the Earth's orbit is much smaller than that of the sun's orbit around our host galaxy. Thus the latter shows a greater anomalous velocity.

That's only for spherical concentric shells. If, for instance, a star passed within 2 AU of the Sun, it would definitely affect the Earth's orbit, even though all the mass is outside the Earth's orbit.

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>The weirdness is that dark matter doesn't interact electromagnetically (although neutrinos don't do that either)

How do we know that Dark Matter is not just a lot of free floating neutrinos? This seems like too obvious an answer so I assume we have a reason.

EDIT: Never mind, I found it on Google.
What rules neutrinos out of the running for dark matter is that in the Standard Model, they are considered “hot” particles, meaning they travel at speeds close to the speed of light. For a particle to constitute dark matter, it must be “cold,” or travel slowly compared to light.

Neutrinos are too light, and move too quickly, so they wouldn't collapse into halos of the right size.

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So dark matter halos in galaxies are formed as a result of cosmological expansion?

Assuming dark matter wasn't created cold enough to cluster on those scales, then yeah, that's a fair interpretation. However it should be noted that on galactic scales, the temperature of the dark matter was not the limiting influence on when structures began to form. That was set by the initial amplitude of variations in the density of the universe and the details of how they grew over time (which are determined by certain aspects of the history of the universe).

Since the dark matter temperature was not the limiting influence, we actually have no clear evidence what its temperature initially was. However, by probing dark matter halos at smaller and smaller scales, we might be able to determine it. The impact of temperature becomes more important at smaller scales.

I see, that's very interesting! So the only 'dissipative' force on dark matter is the cosmological expansion, and the resulting clustering is determined by the density fluctuations of the early universe?