Why is $g$ tied to the oscillator strength $f$ in $\log{gf}_{\odot}$ The $\log{gf}$ value of an element in a star, where $f$ is the oscillator strength, or probability of transition in an atom, is a scientifically pursued quantity with relation to the model atmosphere of the star. Why is the oscillator strength tied to $g$? And what is the motivation for including this quantity in a stellar model?
You don't make it clear, but you may be confused about what $g$ is. It is the statistical weight of an atomic energy level.

The $gf$ value, which refers to a particular transition between two energy levels in an atom/ion, is used because there is symmetry in terms of emission/absorption processes once the statistical weight is taken into account.
$$g_1 f_{12} = - g_2 f_{21}$$

If you just quoted $f$, then you would also need to know what the appropriate statistical weight was in order to calculate transition probabilities.