Are the women in Shakespeare's plays actually more powerful than the men? Do they have more power than the men want us to think? Why?
Short answer: no (or at least, it depends on the woman), and not really.
Slightly longer answer: Some women in some plays are more powerful than the men. However, look at the major actions taken in the major plays—major for good and ill. Look at Iago and all that he does, Othello, Prospero in The Tempest, Hamlet, and all the men in Romeo and Juliet. Even there, it is Lord Capulet forbidding and allowing things.
Now, that said, some women in Shakespearean plays have tremendous power (Lady Macbeth, for example), and in many cases where they have no overt power, they have much ability to woo, distract, or charm. They often have considerable feminine or intellectual power. Men have legal, physical, and cultural power.
As far as what the men want us to think…it isn't a matter of what the men want others to think. It's what Shakespeare and his society believed.