Lower bound on positive coefficients of the optimum of 0,1-linear programming problem
$\frac 1n$ is not a lower bound.
For example, let $n=10$.
Consider 3 set of equations (you can get equalities by considering both $a^\top v \le 1$ and $a^\top v \ge 1$):

\begin{align}
v_1 + v_2 + v_3 \qquad = 1\\
\qquad v_2 + v_3 + v_4 = 1\\
v_1 \qquad  + v_3 + v_4 = 1\\
v_1 + v_2 \qquad + v_4 = 1\\
\end{align}
It's easy to check that $v_1=v_2=v_3=v_4=\frac 13$ is the unique solution.

Consider $v_5 + v_6 + v_7 + v_8 + 0 \cdot v_9 = 1$ and, similarly to the above case, all cyclic shifts of $(v_5, \ldots, v_9)$. Again, the unique solution is $v_5 = v_6 = v_7 = v_8 = v_9 = \frac 14$.

The last equation is
$$v_1 + v_2 \qquad + v_5 \qquad + v_{10} = 1$$
Then $v_{10} = 1 - 2 \cdot \frac 13 - \frac 14 = \frac 1 {12}$.


Generalizing it, you can always obtain coefficient $\frac 1{k (k+1)}$ using $2k + 4$ variables.
I think that by properly selecting numbers $p_1, \ldots, p_k$, you should be able to get a coefficient as small as $O\left(\frac 1 {\prod_i p_i}\right)$ with $O\left(\sum_i p_i\right)$ variables.