Most of the observed precession (about 5557 arcsecs/cent) is simply due to precession of the equinox, and another 532 arcsecs/cent is due to the pulls exerted by the other planets. These effects would be virtually identical for both Newtonian gravity and general relativity. The remaining 43 arcsecs/cent has no explanation within Newtonian gravity, but in general relativity this extra precession is a natural feature of a single test particle orbiting in a spherical field, not related to the precession of the equinox or the pull of the other planets. When you set up the equations to determine the magnitude of this effect, you omit the precession of the equinox and the pull of the other planets. That's why you get just the extra 43 arcsec/cent. You could do the calculation for the whole effect, but it's much more complicated.