The usual ideal gas model is that collisions are perfectly elastic, so even if you do factor in collisions they don't actually change anything.

They don't change ANYTHING? Suppose I start with a gas of molecules all moving at the same speed but in different directions, and they have elastic collisions off the walls of the volume. If they do not collide with each other, they never "thermalize," their speeds stay the same forever as they bounce off the walls but not off each other. But if they do bounce off each other, the velocity distribution does become thermalized by their collisions, even when these collisions are elastic. So collisions don't chage ANYTHING? They change the distribution of velocities to a thermal one, which seems to me to be something.

The ideal gas approximation should be quite close to the actual value for gases like Helium.

So even if an ideal gas maintained perfect decorrelation between molecule positions in an ideal gas with collisions, which I do not think you can demonstrate (and appealing to an unlinked sequence does not count as a demonstration), you would still have to face the fact that an actual gas like Helium would be "quite close" to uncorrelated, which is another way of saying... correlated.