I love this question! I don't know if it's ever possible to know "why"an author chooses a character to serve as the main character, the protagonist. It is possible that the dilemmas experienced by women when it comes to love, were of great interest to Shakespeare. Without question, the problems surrounding love were central for Shakespeare's comedies, since structurally, Comedy was defined as ending in at least one marriage.
Women, as we know, had very little actual power when it came to deciding the most important aspect of their life--to whom they would be married. Of course, Shakespeare was writing in a moment of great change culturally, and it wouldn't be long before marriage for love replaced marriage for family alliance in English-speaking cultures.
Tragedy, as a dramatic form, came out of a Greek tradition in which the tragic hero was meant to be a very high born man of great power and influence--in short, the kind of regal and prideful person ripe for a fall. It would take a bit longer in dramatic history for tragedy to become possible for more simple, regular humans.
For Shakespeare, the hero needed to be high born and important, if not royal. And, ironically for an age with perhaps the world's most famous and powerful Queen, in Shakespeare's day, for the most part, important and powerful equalled male.
Great question and answers.
In the tragedies, women were the victims, for the most part, of the men and their bloody, high stakes games and power struggles.
In the comedies, Shakespeare could show intelligent and amazing women. Viola is playful with Orsino but there is a tinge of sadness that he detects. Cesario is indeed wise about women. She teaches this man who is in love with love in the beginning of the play and he becomes a man who has fallen in love with an extraordinary youth (Viola).
Portia is another extraordinary woman. She is beautiful, she is intelligent and she is filthy rich. No wonder men flock to attempt to find her portrait in one of the caskets devised by her father. After "winning" her, Bassanio fails his first test with the ring. He learns not to give away something valuable like the ring (a symbol of their love) lightly, even if begged for by a brilliant young attorney who has just saved his best friend's life.
Beatrice and Hero teach the men, especially Claudio and Don Pedro, not to believe everything they see or hear, particularly if it involves Don John or any of his men.
In the comedies, Shakespeare gives voice to the positive power of women. His females in the comedies become teachers. They teach the men in their lives about being gentle and human and compassionate and many other valuable lessons.
Maybe, Shakespeare, in his tragedies, showed the male as the protagonist because the male world is more violent so in his balanced world, (the other side of the coin, so to speak), he portrayed women as the protagonists in his comedies because their world is gentler and more romantic.
Shakespeare's comedies deal with love and romance, while his tragedies are tales of ambition, power, bloodshed and revenge. Young and energetic women with a lot of passion fit well into the framework of comedies, whereas strong and ambitious men with complications of head and heart people the world of tragedies.
Take, for example, a comedy like Twelfth Night or As You Like It where we find women such as Viola, Olivia and Rosalind who dominate the dramatic action and impress us with their exemplary beauty of body and mind. They come up as clear winners over their male counterparts. On the other hand, plays like Macbeth, Hamlet, King Lear and Othello are dominated by highly striking and exceptional men who suffer because of their own characteristic flaws. Even these tragedies contain very remarkable women, such as, Lady Macbeth, Ophelia, Cordelia and Desdemona; but all of them are veritable victims of viciousness as embodied by men: Macbeth's ambition, Hamlet's doubt and dilemma, Lear's anger and Othello's jealousy. In the comedies loving, caring women are winners in their run of life, whereas in the tragedies, the women are great losers and victims.