Although I have never thought of the play in those terms, I can see the validity of the question.
Leontes's unreasonable behavior proves to be the death (winter) of his relationship with his wife. Death is also literally there with the death of Mamillius as a result of Leontes's actions plus the death of Antigonus is also a result of his madness.
When we jump ahead in time to Bohemia, it is also spring, a time for new life, awakenings, and new beginnings. It is springtime both literally and figuratively. We see Perdita now sixteen, a youth, in the springtime of her life. We see the beginnings of young love between Perdita and Florizel.
The spring of Bohemia breathes new life into Sicily and when what has been lost (Perdita) has been found, Leontes is in a sense reborn. This new life brings him to a new found reality and the love that was destroyed has been restored through the magic of Paulina, the awakening of Hermione.
Summer and fall do not come into play but Shakespeare has pitted the ideas of winter (death) against spring (life) successfully.
Shakespeare might also have been influenced by Celtic folklore. Part of the celebration of Samhain is the battle between the summer king and the winter king with the winter king winning.
The beauty of Shakespeare is that he gives us so much to think about and he can be interpreted in many ways.