Of course, the gravitational field of the rest of observable universe is also present, but that is an extremely UNIFORM EXTERNAL gravitational field over the Solar System that it has NO effect on the INTERNAL motions of the Solar System.

In fact, whenever one describes a particular gravitational field as on or over some particular system of astro-bodies, it is always understood that gravitational field of the rest of observable universe is also present, but that gravitational field is an extremely UNIFORM EXTERNAL gravitational field over the particular system, and so has NO effect on the INTERNAL motions of the particular system.

The center of mass of the Earth would then be the origin of an ideal inertial frame with coordinate axes attached to the unrotating Earth. The inertial frame can also be described as the center-of-mass (CM) inertial frame of the Earth.

In this counterfactual case, every point on the surface of the Earth is NOT accelerated with respect to the ideal inertial frame attached to the Earth's center of mass, and so also defines an ideal LOCAL inertial frame (i.e., an inertial frame at the point and nearby surroundings).

You do NOT need for the point to be the center of mass of any object and usually you would NOT call such Earth surface inertial frames CM inertial frames even if they were actually CM inertial frames.

For an example of a picturesque piece of Earth surface (but NOT an ideal one), see the figure below (local link / general link: alpine_tundra.html).

1. The gravitational field of the Solar System is NOT exactly UNIFORM over the Earth. The Moon's gravitational field and secondarily the Sun's gravitational field vary across the Earth. Note the Moon's gravitational field is weaker than the Sun's, but its variation is greater.

   And it is the variation in the gravitational field that causes the tidal force on the Earth (see Mechanics files: tide_earth.html and The Tidal Force and the Earth).

   Note, the tidal force is actually a combination of an force and inertial force (see Mechanics file: newtonian_physics.html: Force Categories).

2. Now the overall gravitational field of the Moon pulls Earth into orbit around the Earth-Moon system center of mass and the gravitational field the Sun pulls the orbit around the Sun (more exactly the Solar System center of mass).
   Note a bound orbit is a rotation relative to the observable universe of an astro-body about a center of mass for the gravitationally bound system containing the astro-body. Such a gravitationally bound system, of course, constitutes a CM inertial frame.

3. The tidal forces of the Moon and Sun stretch the Earth along the lines to, respectively, the Moon and Sun.

   The stretching causes the Earth tides (i.e., the water tide) and also the Earth's land tide and atmospheric tide.

   Now the tidal forces of the Moon and the Sun have virtually NO effect on small-scale everyday life and small-scale laboratory experimentation, and so can be neglected for most purposes, but NOT all purposes as discussed below.

   For more on the tides, see Mechanics files: tide_earth.html and frame_basics.html: The Tidal Force and the Earth.

1. The Earth actually has the Earth's daily axial rotation. This means that every point on the surface of the Earth is in acceleration.

2. Now, in principle, for all calculations you could just use the inertial frame defined by the Earth's center of mass as an origin with coordinate axes unrotating relative to the observable universe. For brevity, let's call this inertial frame, the UNROTATING FRAME.
   Note that unless one says rotating frame, one usually means an inertial frame NOT rotating relative to the observable universe.

   In fact, the UNROTATING FRAME is SUPER INCONVENIENT for all purposes since almost all Earthly things (including us) mostly rotate with the Earth.

   Therefore, for Earthly purposes, we use a rotating frame that rotates with the Earth and use rotating frame inertial forces to convert said rotating frame in an inertial frame.

3. The rotating frame inertial forces are the centrifugal force, the Coriolis force, and the Euler force. The Euler force is for accelerating rotation and is NOT needed for the case of the Earth.

   We explicate the conversion and the needed rotating frame inertial forces in frame_basics.html: Rotating Frames and the Centrifugal Force and the Coriolis Force.

   But the acceleration of the surface of the Earth is actually so low (⪅ 0.03 m/s**2: see Wikipedia: Gravity of Earth: Latitude) that for most small-scale purposes (but NOT all purposes), you can treat every surface point as defining a LOCAL inertial frame (i.e., an inertial frame at the point and nearby surroundings) without using the centrifugal force and the Coriolis force.

   How small is small scale? Small-scale everyday life and small-scale laboratory experimentation.

4. When do you need to use the centrifugal force and the Coriolis force? For respectively, the shape of of the Earth and weather.

5. For more on these uses and rotating frames in general, see Mechanics files: frame_basics.html: Rotating Frames and the Centrifugal Force and the Coriolis Force, frame_basics.html: Rotating Frames Explicated, frame_basics.html: The Centrifugal Force of the Earth's Rotation, and frame_basics.html: The Coriolis Force of the Earth's Rotation.