  In this paper, we have outlined the development of an autonomous dynamical
system within a general scalar-tensor gravity framework. This framework
encompasses the overall structure of the non-minimally coupled scalar field
functions for both the torsion scalar ($T$) and the boundary term ($B$). We
have examined three well-motivated forms of potential functions and constrained
the model parameters through dynamical system analysis. This analysis has
played a crucial role in identifying cosmologically viable models. We have
analysed the behaviour of dynamical parameters such as equation-of-state
parameters for dark energy and the total, as well as all the standard density
parameters for radiation, matter, and dark energy to assess their compatibility
with current observational data. The phase space diagrams are presented to
support the stability conditions of the corresponding critical points. The
Universe is apparent in its late-time cosmic acceleration phase via the dark
energy-dominated critical points. Additionally, we compare our findings with
the most prevailing $\Lambda$CDM model. The outcomes are further inspected
using the cosmological data sets of Supernovae Ia and the Hubble rate H(z).
