In this study, we investigate the flavor-decomposed gravitational form factors(GFFs) of the nucleon and the baryon octet in a mean-field approach based on the large $N_{c}$ limit of QCD. Our central focus lies on illuminating the distinctive role played by the separate quarks in this context. We first scrutinize the behavior of the gravitational form factors in the large $N_c$ limit, tracing the relations governed by spin-flavor symmetry within the framework of flavor SU(3). The results are compared with those in flavor SU(2). Additionally, we quantitatively assess the fraction of light-front momentum carried by individual quark flavors within the nucleon. Notably, we unravel variations arising from the mass distribution, as evidenced through the non-conserved form factor of the nucleon, known as the cosmological constant-term $\bar{c}(q^2)$. Venturing further, we explore not only the decomposition of the total angular momentum into the orbital angular momentum and intrinsic spin, but also its flavor decomposition. Furthermore, we delve into the intricate interplay between the $D$-term and $\bar{c}$ form factors, discerning their collaborative impact on the stabilization of the nucleon system. Alongside this, we undertake an analysis, questioning the assumption of "large $N_{c}$ blindness'' concerning $D^{u-d} \sim 0$. Our examination concludes that such an assumption finds justification predominantly within the framework of flavor SU(3) symmetry. Using the spin-flavor symmetry, the baryon octet GFFs can be easily obtained, and various sum rules between the baryon octet GFFs are discussed. Finally, we introduce the generalized electromagnetic form factors, where the flavor structure of the electromagnetic current is induced into the GFFs. Similar to the electromagnetic form factors, we find that these newly introduced form factors possess the $U$-spin symmetry.