The main point is that:
On one hand, ${\bf r}^{\prime}(t)$ is an dynamical active variable of the system. Its time evolution is determined by the equations of motion of the system, i.e. Newton's 2nd law. Here ${\bf v}^{\prime}:=\dot{\bf r}^{\prime}$.
On the other hand, ${\bf R}(t)$ is a passive (background) variable with pre-defined time-dependence. Here ${\bf V}:=\dot{\bf R}$.
We could in principle define e.g.
$$\tag{1} F_1~:=~\frac{m}{2}\int_0^t \! dt^{\prime}~
\dot{\bf r}^{\prime}(t^{\prime})^2,\qquad
\dot{F}_1~=~\frac{m}{2}\dot{\bf r}^{\prime 2}, $$
$$\tag{2} F_2~:=~\frac{m}{2}\int_0^t\! dt^{\prime}~
\dot{\bf r}^{\prime}(t^{\prime})\cdot \dot{\bf R}(t^{\prime}),\qquad
\dot{F}_2~=~\frac{m}{2}\dot{\bf r}^{\prime}\cdot\dot{\bf R} ,$$
$$\tag{3} F_3~:=~\frac{m}{2}\int_0^t \! dt^{\prime}~
\dot{\bf R}(t^{\prime})^2,\qquad
\dot{F}_3~=~\frac{m}{2}\dot{\bf R}^2. $$
The problem with e.g. the definition (1) is that $F_1$ would be a non-local quantity. It would depend on the dynamical active variable ${\bf r}(t^{\prime})$ in the past $t'<t$. See also this Phys.SE post.
On the other hand, there is no problem with locality in the definition (3) of $F_3$ because the passive variable ${\bf R}(t)$ is pre-defined for all times $t$. In other words, $F_3$ is just another passive background variable.
References:
- L.D. Landau & E.M. Lifshitz, Mechanics, vol. 1, (1976), p. 127.
