Dark matter's effect on Solar System scales exists in theory, but it is nigh undetectably small.
A related question deals with the density of dark matter in the Solar System. It comes out to less than one hydrogen atom per cubic centimeter, some 20 orders of magnitude less dense than a typical solid or liquid, and 17 orders of magnitude less dense than air. Moreover, this is a rather uniform distribution of matter, so its gravitational effect on planets' motions is essentially nothing.
Once you zoom out to the scale of the galaxy, you start to see that dark matter is more concentrated toward the center of the galaxy and less concentrated (very) far out. Thus it adds to the gravitational potential well of the galaxy, affecting the orbits of stars and star clusters going around the galaxy.
So it comes down to the fact that the dynamics of the Solar System are utterly dominated by an extremely concentrated overdensity (the Sun, as well as the planets) compared to the average density of the galaxy (a few atoms per cubic centimeter). This is an important point, by the way: Being on a planet near a star is a highly unlikely place to find oneself compared to typical locations drawn uniformly from the universe; most places in space are much, much more devoid of normal matter.
One analogy is to think of a desert with gently rolling sand dunes. On large scales, the shape of the landscape is well approximated by smooth curves capturing the shape of the dunes. But if you look at the sand with a microscope, the surface is very rough, with all sorts of little bits of rock piled haphazardly. A microscopic organism crawling around on the sand hardly notices the shape of the dune it's on, but it cares very much about the size of a nearby sand grain.
Also note there is a lot of other evidence for dark matter beyond the dynamics of galaxies, and all this evidence agrees in terms of how much dark matter is needed to explain the observations. In fact, I'd say we would all be as incredulous as you sound if galactic dynamics were all we had. But the success of cosmology (for which no one has developed a credible dark-matter-free alternative) is strong evidence that dark matter is indeed spread out in the universe.