According to Newton:
$r(t+dt)=r(t)+v(t)dt$
and
$\dot{r}(t+dt)=\dot{r}(t)+\frac{F(t)}{m}dt$.
As you can see, there is no freedom to choose the trajectory - it is determined with the instant values of force and velocity. "Future" is determined with "present".
A particle never "chooses" the optimal trajectory to go from a known position in the past $r(t_1)$ to a known position in the future $r(t_2)$. The future data are not involved in the dynamics. But the least action "principle", apart from good equations, is based on the future data $r(t_2)$ which is mathematically possible but physically meaningless.
There is no a "least action principle" proceeding only from the initial data. Instead, the Newton equations with the initial data suffice to solve physical problems ;-)