If spontaneous symmetry breaking only occurs in infinite systems, why do we observe similar effects in finite systems?

This question exactly is addressed and rigorously treated by N.P. Landsman. The explanation is that in the large $N$ limit the symmetric ground state becomes exponentially sensitive to asymmetric perturbations, while the first excited states, although unstable, become very close in energy to the symmetric state and decay exponentially slowly in any direction, thus the system dynamically finds itself in a symmetry broken state already at a finite by very large $N$.

Not sure if this matches your puzzle, but: when you write something like $[\rho, \mathcal{G}]$, you are thinking about measuring the symmetry existence by looking at the expectation value of the operator itself. However, in completely quantum sense, you should define the symmetry with the expectation of correlators other than operators themselves.

You might find the note of Lec.1 here helpful, which used transverse Ising model as an example, and the SB definition is mentioned on Pg.9

https://learning-modules.mit.edu/materials/index.html?uuid=/course/8/fa17/8.513#materials