We present atomistic computations within an empirical pseudopotential framework for the electron s-shell ground state g tensor of InGaAs quantum dots (QDs) embedded to host matrices that grant electronic confinement. A large structural set consisting of geometry, size, and molar fraction variations is worked out which also includes a few representative uniform strain cases. The tensor components are observed to display insignificant discrepancies even for the highly anisotropic shapes. The family of g-factor curves associated with these parameter combinations coalesces to a single universal one when plotted as a function of the gap energy, thus confirming a recent assertion reached under much restrictive conditions. Our work extends its validity to alloy QDs with various shapes and finite confinement that allows for penetration to the host matrix, placing it on a more realistic basis. Accordingly, the electrons in InGaAs QDs having s-shell transition energies close to 1.13 eV will be least susceptible to magnetic field. We also show that low indium concentration offers limited g-factor tunability under shape or confinement variations. These findings can be taken into consideration in the fabrication and the use of InGaAs QDs with g-near-zero or other targeted g values for spintronic or electron spin resonance-based direct quantum logic applications.