Shakespeare's women were much a product of their times, as was he.
I think that Shakespeare uses the misogyny in his age provocatively, though. It is difficult to determine what Shakespeare himself actually thought. He was a highly successful playrite and knew how to stir up an audience. Much ink has been spilt trying to interpret his meaning--scholars will look, for instance, at which characters speak which lines as evidence of what he thought, with the notion that the words of the most trustworthy characters are the ones most likely to mirror the author's own thoughts. But that is slippery scholarship.
I think that part of Shakespeare's power to appeal to such a varied audience over such a long time span is that though his characters are often didactic, his plays are pointedly not. Hence, a character can say something utterly misogynistic (Iago, in Othello, for instance) and patriarchal (Emilia, Iago's wife) and the play itself may even close with an apparently instructive note, yet the ideas sown throughout the play may remain ripe for discussion. Given the nature of the times, and the nature of the business Shakespeare is in (entertainment), it would be difficult to claim anything other than the fact that he provoked us. Yet perhaps the fact that he exposed these ideas and, by doing so, challenged his audience to think about them, suggests that he thought about them some.
The harshest example of such bias is the treatment of Hero. She is held to a sex-specific standard and then slandered for it; a man would not have been treated that way. She essentially collapses due to mistreatment.
Though it is well-meant, Benedict's defense of Hero is also patriarchal. He and the surrounding society assume a woman needs a man to protect her.
Finally, even Claudio's willingness to make amends by marrying Antonio's daughter (who is masked) is patriarchal; it assumes the right of men to dispose of women, showing their relative lack of power in this society.
As for Shakespeare "embodying" such a bias, I'd have to say yes, he did—but so did almost men and women of the period, and Shakespeare's was less than most, given the power he gave to female characters.