I’m somewhat surprised to find that Newton talked about this issue, writing Page:Newton's Principia (1846).djvu/537 - Wikisource, the free online library
“About the sun thus librated the other planets are revolved in elliptic orbits (p 403), and, by radii drawn to the sun, describe areas nearly proportional to the times, as is explained in Prop. LXV. If the sun was quiescent, and the other planets did not act mutually one upon another, their orbits would be elliptic, and the areas exactly proportional to the times (by Prop. XI, and Cor. 1, Prop. XIII). But the actions of the planets among themselves, compared with the actions of the sun on the planets, are of no moment, and produce no sensible errors. And those errors are less in revolutions about the sun agitated in the manner but now described than if those revolutions were made about the sun quiescent (by Prop. LXVI, and Cor. Prop. LXVIII), especially if the focus of every orbit is placed in the common centre of gravity of all the lower included planets; viz., the focus of the orbit of Mercury in the centre of the sun; the focus of the orbit of Venus in the common centre of gravity of Mercury and the sun; the focus of the orbit of the earth in the common centre of gravity of Venus, Mercury, and the sun; and so of the rest. And by this means the foci of the orbits of all the planets, except Saturn, will not be sensibly removed from the centre of the sun, nor will the focus of the orbit of Saturn recede sensibly from the common centre of gravity of Jupiter and the sun”
I summarize this as "The deviations in orbits due to the interaction between the planets (the gravity of Earth on Venus, etc.) are tiny; a bigger effect is the fact that the focus of planet A’s orbit is the center of mass of Sun and all the planets inside the orbit of planet A. Since the center of mass of the Sun and all the planets inside Jupiter is essentially the center of mass of the Sun (since all the inner planets are tiny compared to the Sun), only for Saturn do we have to worry about the effect of Jupiter.