LP formulation and integer solution existance
I don't know why you added this in your second constraint:


  ..., and which is a maximal set with that property, ...


I would model it simply as: "For every two agents $x_i$ and $x_j$, if their sets $S_i$ and $S_j$ intersect, then $x_i + x_j \leq 1$"
$$
x_i + x_j \leq 1 \quad \forall (i,j): i \lt j \text{ and } S_i \cap S_j \neq \emptyset$$

Notice that every row in the constraint matrix has exactly two ones, and we can think of it as a bipartite graph, which is TUM.

This matrix is not TUM, and there is no guarantee that it will provide Integer Solutions, as we can see in the counter-example:
$$ S_1 \cap S_2 \neq \emptyset, S_1 \cap S_3 \neq \emptyset, S_2 \cap S_3 \neq \emptyset $$
\begin{bmatrix}
1 &amp; 1 &amp; 0\\
1 &amp; 0 &amp; 1\\
0 &amp; 1 &amp; 1\\
\end{bmatrix}

That doesn't mean that your problem does not have an integer optimal solution, simply that this Linear Formulation will not always yield an integer solution. For example, the solution of your LP might be $x_1 = 0.23848$, which satisfies all the constraints and is the optimal solution, but it is not integer.

If your goal is to solve this problem, and get the Integer Optimal Solution, you can use an Integer Linear Formulation and use any ILP solver to get to that solution.

The Integer Linear Formulation allows constraints like $ x_1 \in \{0, 1\}$, that will ensure that this variable will be either 0 or 1.

The purpose of proving that it is TUM, is to avoid using Integer Linear Formulations, since they are slower to solve than Linear Programs.