Context. The chemical evolution history of slow neutron-capture elements in the Milky Way is still a matter of debate, especially in the metal-poor regime ([Fe/H] < −1). Aims. Based on Gaia-ESO spectroscopic data, a recent study investigated the chemical evolution of neutron-capture elements in the regime [Fe/H] > −1. Here, we aim to complement this study down to [Fe/H] = −3, and focus on Ba, Y, and Sr, along with the abundance ratios of [Ba/Y] and [Sr/Y], which give comprehensive views on s-process nucleosynthesis channels. Methods. We measured the local thermodynamic equilibrium (LTE) and non-local thermodynamic equilibrium (NLTE) abundances of Ba, Y, and Sr in 323 Galactic metal-poor stars using high-resolution optical spectra with high signal-to-noise ratios. We used the spectral fitting code TSFitPy together with 1D model atmospheres, using previously determined LTE and NLTE atmospheric parameters. Results. We find that the NLTE effects are on the order of ∼ − 0.1 to ∼0.2 dex, depending on the element. We find that stars enhanced (deficient) in [Ba/Fe] and [Y/Fe] are also enhanced (deficient) in [Sr/Fe], suggesting a common evolution channel for these three elements. We find that the ratio between heavy and light s-process elements [Ba/Y] varies weakly with [Fe/H] even in the metal-poor regime, which is consistent with the behaviour in the metal-rich regime. The [Ba/Y] scatter at a given metallicity is larger than the abundance measurement uncertainties. Homogeneous chemical evolution models with different yield prescriptions are not able to accurately reproduce the [Ba/Y] scatter in the low-[Fe/H] regime. Adopting the stochastic chemical evolution model by Cescutti & Chiappini allows us to reproduce the observed scatter in the abundance pattern of [Ba/Y] and [Ba/Sr]. Based on our observations, we have ruled out the need for an arbitrary scaling of the r-process contribution, as previously suggested by the authors behind the construction of the model. Conclusions. We show how important it is to properly include NLTE effects when measuring chemical abundances, especially in the metal-poor regime. This work demonstrates that the choice of the Galactic chemical evolution model (stochastic versus one-zone) is key when comparing models to observations. Upcoming large-scale spectroscopic surveys such as 4MOST and WEAVE are poised to deliver high-quality data for many thousands of metal-poor stars and this work gives a typical case study of what could be achieved with such surveys in the future.