Yeah just like the Mesopotamian Goddess Tiamet who was originally a mother goddess of creation but later transformed into a giant serpent or dragon and after being slain by one of their male heroes she gave birth to various creatures from her body including humans (I think, this was based on what I remember from briefly googling her a while back).
Anyways the idea of snakes being phalic symbols to represent sexual arousal is interesting, but also when I think of the umbilical cord connection it's like how in ancient times a snake could both destroy life with its venomous bite but also help give life by having its poison turned into medicine. Kinda like the ancient symbol of ouroboros, the snake or dragon devouring its own tail, destroying and renewing itself at the same time.
There might be some connection between this and the way ancient people viewed their creation goddesses like Gaia or Tiamet? Gaia was so powerful that not only did she give birth to all of creation but she could also destroy her own creations too, and she even tried to destroy the gods themselves for overthrowing the titans who were her other children by giving birth to monsterous giants like Typhon.
And this snake symbolism is like something I think speaks to the role women played in actual childbirth. They could give a new life either through birth or they could lose a life through stillbirth, sometimes losing their own lives in the process. And of course there's also abortion which has been practiced for thousands of years in places like Ancient Egypt for example...
Childbirth was probably a very mysterious thing to a lot of ancient humans back in the day I'm sure.
So yeah, snakes seem perfect as both phalic symbols and as symbols for childbirth. 