I was wondering whether Martin Sleziak's excellent answer to this question could be generalized:

Suppose that $S$ is an infinite set, and let ${\cal L}$ be a collection of subsets of $S$ such that for all $L_1\neq L_2\in {\cal L}$ the intersection $L_1\cap L_2$ contains at most one element. Is there $M\subseteq S$ such that

1. $M$ intersects all the elements of ${\cal L}$, but
2. for all $m\in M$, the set $M\setminus\{m\}$ no longer intersects all the elements of $\cal L$

?

I was wondering whether there is a generalization of Martin Sleziak's excellent answer to this question:

Suppose that $S$ is an infinite set, and let ${\cal L}$ be a collection of subsets of $S$ such that for all $L_1\neq L_2\in {\cal L}$ the intersection $L_1\cap L_2$ contains at most one element. Is there $M\subseteq S$ such that

1. $M$ intersects all the elements of ${\cal L}$, but
2. for all $m\in M$, the set $M\setminus\{m\}$ no longer intersects all the elements of $\cal L$

?