I think this is a very good and deep question that maybe related to Prof.Wen's question, and I would try to answer you in my understanding.
Let's take the nearest-neighbor spin-1/2 antiferromagnetic Heisenberg Model on the square lattice as an example, where the symmetry breaking Neel state only emerges in the thermodynamic limit.
As you have mentioned, note that only when the system has finite size, (i.e., the square lattice constitutes of two sublattices A and B with equal sizes $N_A=N_B$, and hence the total number of spins $N=N_A+N_B$ is even), the ground state is unique and is exact a singlet state with $SU(2)$ spin-rotation symmetry (Marshall,1955; Lieb and Mattis, 1962). However, as the system size becomes large, there are many low-lying excited states with very small energy gap $\Delta$ above the singlet ground state, and these low-lying states break the $SU(2)$ spin-rotation symmetry(i.e., they may be triplet states). More subtlely, as $N$ approaches $\infty $, those nearly degenerate ground states would 'collapse' into the ground state in the thermodynamic limit ($\Delta\rightarrow 0$), indicating that the Neel state is in fact a superposition of many nearly degenerate ground states in the thermodynamic limit. Thus, in the strict thermodynamic limit, there exists an $SU(2)$ symmetry breaking state of the 'highly' degenerate ground states.
Indeed, this is a nontrivial example of spontaneous symmetry breaking since the exact ground state of the finite system does not break $SU(2)$ spin-symmetry while there emerges spontaneous symmetry breaking (due to the nearly degenerate ground states) in the thermodynamic limit. The above argument is just a very rough picture, and I also feel it is somewhat difficult to understand how the ground state degeneracy happens for a gapless system in the thermodynamic limit? Moreover, I also get another question: Theoretically, as there are 'highly' degenerate ground states containing both $SU(2)$ symmetric singlet state and symmtery-breaking Neel states in the thermodynamic limit, why we are used to saying the ground state of the antiferromagnetic Heisenberg Model on the square lattice is a Neel state rather than a singlet state?