In the case of a 3-body solar system, they wouldn't. But as soon as one of these mass dots tries to land (or stand) on one of the other mass dots, you'd see a difference.

If you say that this violates the assumptions of the basic model, you're right. But if you look at all the craters on the moon, the model assumptions are violated pretty often - the trick is recognizing when you need a different model, which is always easy in retrospect, but can be difficult to anticipate.

Yep, that's true. I just thought that is was interesting that the shape of a planet doesn't affect its trajectory under the laws of gravity alone (i.e. no collisions etc). It's also true that physicists do use explicit approximations often, for example in statistical mechanics, or by using a linear approximation instead of an exact quantity. That's similar to calculating the asymptotic running time of an algorithm instead of the exact number of steps it takes (except the limit in the physics case takes some quantity to zero to get a linear approximation whereas for the algorithm we take the limit of the input data size to infinity).

...hm, how many orders of magnitude down are the tidal forces this ignores?

Do tidal forces have any effect on the center-of-mass trajectories of planets?

They should, they transfer energy between rotational and orbital velocities. For example the earth is gradually spinning more slowly (as can be seen from daily/annual growth rings in corals), while the moon is orbiting faster and getting farther away.