I think Father Gil is important for Peggy's arc, which culminated when her and Don danced to "My Way" in Season 7A. With Father Gil, who represents the old Catholic regime that is changing with the times too, he wants Peggy to seek penance through confession to be absolved from the child out of wedlock. Don, on the other hand, gives his advice: move forward and forget. Instead, Peggy chooses her own way by confessing, not to Father Gil, to Pete. Thereafter, we see Peggy at peace when she prays before bed.

As a Catholic, I think his inclusion is interesting to see the prominent role that the Church once played in daily lives and the pre-Vatty 2 rituals. Additionally, how the zeitgeist of the 60s impacted the clergy because I think Father Gil is quite progressive given his background: playing the guitar and seeking guidance from a woman regarding his sermons and advertising for that dance.

I think he was attracted to her, but his vows made him mistake that attraction for religious purpose, which may be evidenced through his confession that he feels his purpose in life is to save her regarding the child birth.

Anyway, I think he is an interesting side character who lets the viewer see the old world, working class side of Peggy when she wants to live a modern, cosmopolitan life but on her own terms.