Why were women really needed in Shakespeare's plays?
If plays are a reflection of life and the people who live life, then women are essential. Shakespeare was a great observer of the human condition. Women are a part of that.
In his comedies, the women are important because he is able to show us strong, intelligent, and independent women. Most of the male characters don't truly understand what love is until they meet one of these women (often disguised as males).
Many of Shakespeare's plays deal with love and how can you have a Romeo without a Juliet? Where would the great love of Antony be without Cleopatra?
Where would the play King Lear be without his daughters. Father and son relationships are quite different from father daughter relationships. Shakespeare had two daughters.
I could go on but the answer should be obvious. Women make up more than half of the population. They are lovers, wives, mothers, daughters, sisters, etc. Shakespeare gave us all of these.