It is impossible for the Earth to be the center of mass and be stationary. The planets move; therefore the center of mass of the solar system moves. Note that the center of mass is
not the point at which all gravitational forces cancel out. For example, the Sun-Jupiter system (from the first reference below):
The L1 point is the point where the Sun's and Jupiter's gravity cancel out. Far from the center of mass.
References:
As the Universe TurnsDialogue on the Center of Mass of the Universe, Part 1Dialogue on the Center of Mass of the Universe, Part 2
Claims that Mach's principle and the gravitational force of the rest of the universe overpower the gravity within the solar system must be accompanied by calculations. That which can be asserted without evidence, can be dismissed without evidence. — Christopher Hitchens.
But we do not live in a two body universe; wherefore Newton's law of two bodies cannot accurately measure reality.
No model can measure reality absolutely accurately. But Newton's laws (which apply to any number of bodies, whether or not an analytic solution can be obtained) measure reality accurately enough for the masses and velocities found in the Solar system. If you want to
demonstrate that Newton's laws are not sufficiently accurate, show us the calculations.
As if you could.