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In a Souslin tree, every antichain is countable, and hence bounded in the tree. Thus, every maximal antichain is refined by a level of the tree. Furthermore, for antichains in $M$, this level by elementarity will be an ordinal in $M$. Since any branch through the tree of height above $\omega_1\cap M$ goes through all such levels, it will therefore meet every such maximal antichain, and hence it will be $M$-generic

One key fact for this argument was the fact that the forcing was ccc---the antichains were countable---but a powerful generalization of similar ideas is the notion of proper forcing, which can be defined in terms of finding conditions that force $M$-genericity for suitable countable elementary submodels.

In a Souslin tree, every antichain is countable and hence bounded in the tree. Thus, every maximal antichain is refined by a level of the tree. For this reason, a cofinal branch through a Souslin tree (found anywhere) is the same thing as a generic branch.

In your case, you have the countable elementary substructure $M$, and want to know about $M$-genericity. But the same observation works. Every antichain in $M$ is refined by a level of the tree in $M$, and since any branch of the tree of height above $\omega_1\cap M$ goes through all such levels, it will therefore meet every such maximal antichain in $M$, and consequently it will be $M$-generic

One key fact for this argument was the fact that the forcing was ccc---the antichains were countable---but a powerful generalization of similar ideas is the notion of proper forcing, which can be defined in terms of finding conditions that force $M$-genericity for suitable countable elementary submodels.

In a Souslin tree, every antichain is countable, and hence bounded in the tree. Thus, every maximal antichain is refined by a level of the tree. Furthermore, for antichains in $M$, this level by elementarity will be an ordinal in $M$. Since any branch through the tree of height above $\omega_1\cap M$ goes through all such levels, it will therefore meet every such maximal antichain, and hence it will be $M$-generic.

In a Souslin tree, every antichain is countable, and hence bounded in the tree. Thus, every maximal antichain is refined by a level of the tree. Furthermore, for antichains in $M$, this level by elementarity will be an ordinal in $M$. Since any branch through the tree of height above $\omega_1\cap M$ goes through all such levels, it will therefore meet every such maximal antichain, and hence it will be $M$-generic

One key fact for this argument was the fact that the forcing was ccc---the antichains were countable---but a powerful generalization of similar ideas is the notion of proper forcing, which can be defined in terms of finding conditions that force $M$-genericity for suitable countable elementary submodels.

In a Souslin tree, every antichain is countable, and hence bounded in the tree. Thus, every maximal antichain is refined by a level of the tree, and since any $\omega_1$-branch goes through all levels, it follows that every such branch meets every maximal antichain, and hence is generic.

In a Souslin tree, every antichain is countable, and hence bounded in the tree. Thus, every maximal antichain is refined by a level of the tree. Furthermore, for antichains in $M$, this level by elementarity will be an ordinal in $M$. Since any branch through the tree of height above $\omega_1\cap M$ goes through all such levels, it will therefore meet every such maximal antichain, and hence it will be $M$-generic.