In many physics problems of interest, the applied force of the $\mathbf{F}_i^{(a)}$ of the $i$'th point particle is not zero.

But your focus should not be on the applied forces $\mathbf{F}_i^{(a)}$. They are there and properly accounted for. Your focus should instead be on the other forces $\mathbf{f}_i$.

The non-trivial statement in eq. (1.43) (apart from Newton's 2nd law for statics $\mathbf{F}_i=0$) is that the other forces produce no virtual work, $$\sum_i \mathbf{f}_i \cdot \delta \mathbf{r}_i ~=~ 0,$$ which e.g. is not true if the other forces include sliding friction.

See also this related Phys.SE post.

In many physics problems of interest, the applied force $\mathbf{F}_i^{(a)}$ of the $i$'th point particle is not zero.

But your focus should not be on the applied forces $\mathbf{F}_i^{(a)}$. They are there and properly accounted for. Your focus should instead be on the other forces $\mathbf{f}_i$.

The non-trivial statement in eq. (1.43) (apart from Newton's 2nd law for statics $\mathbf{F}_i=0$) is that the other forces produce no virtual work, $$\sum_i \mathbf{f}_i \cdot \delta \mathbf{r}_i ~=~ 0,$$ which e.g. is not true if the other forces include sliding friction.

See also this related Phys.SE post.