What is an efficient way to calculate the biggest system of disjunct sets?
This answer assumes that the sets in $\tilde{\mathcal{M}}$ have to be pairwise disjoint (as currently stated, you are requiring them to have empty intersection, which is a monotone condition).

The decision version of your problem is NP-complete, by reduction from independent set. Given an graph $G = (V,E)$, construct a set system with a set $S_x$ for each vertex $x \in V$, and an element $e$ for each edge $e \in E$. The set $S_x$ contains all edges incident to $x$. A family of sets is pairwise disjoint iff the corresponding vertices form an independent set.

We can also go in the other direction, following one of your suggestions. Given a set system, construct a graph in which each vertex corresponds to a set, and two vertices are connected if the sets intersect. A solution to your problem is the same as an independent set in the graph.