One should also be careful, which external force you consider. All you discuss here is about the motion in the homogeneous gravitational field close to Earth.
To that end consider the body under consideration (it can be an elastic or rigid solid body or even a fluid) as a discrete set of point particles, held together by interaction forces, which can be described as "pair forces", i.e., on the particle with the label ##j## the total force is given by
$$\vec{F}_j=\sum_{k \neq j} \vec{F}_{jk} + m_j \vec{g},$$
where ##\vec{g}## is the constant gravitational acceleration on due to the Earth (an approximation valid for motions close to Earth). The equations of motion for the particles thus read
$$m_j \ddot{\vec{x}}_j = \sum_{k \neq j} \vec{F}_{jk} + m_j \vec{g}.$$
Now we sum this equation over ##j## and use Newton's Third Law ("lex tertia"), according to which ##\vec{F}_{jk}=-\vec{F}_{kj}##, i.e., if the interaction force on particle ##j## due to the presence of particle ##k## is ##\vec{F}_{jk}## then the interaction force on particle ##k## due to the presence of particle ##j## is opposite. This means that by summing over ##j## all the interaction forces cancel and you get
$$\sum_j m_j \ddot{\vec{x}}_j=M \ddot{\vec{x}}_{\text{cm}} =M \vec{g},$$
where
$$M=\sum_j m_j, \quad \vec{x}_{\text{cm}}=\frac{1}{M} \sum_j m_j \vec{x}_j.$$
Our calculation shows that indeed the center of mass moves like a point particle in the gravitational field of the Earth,
$$\ddot{\vec{x}}_{\text{cm}}=\vec{g}=\text{const}.$$
For all other forces, and even the gravitational interaction with the Earth when considering the position dependence, is not as simple, because then you have (treating the Earth as a "point particle" sitting fixed at the origin of the coordinate frame),
$$m \ddot{\vec{x}}_j = \sum_{k \neq j} \vec{F}_{jk} -\gamma m_j m_{\text{Earth}} \frac{\vec{x}_j}{r_j^3} \quad \text{with} \quad r_j=|\vec{x}_j|.$$
Then summing over ##j## gives
$$M \ddot{\vec{x}}_{\text{cm}} = -\gamma m_{\text{Earth}} \sum_j \frac{m_j \vec{x}_j}{r_j^3}=\vec{F}_{\text{ext}},$$
and ##\vec{F}_{\text{ext}}## cannot be expressed in terms of ##\vec{x}_{\text{cm}}##!
