@Purplelibraryguy:
To be honest, the satyrical interpretation isn't mine, but from several scholars quoted in the Wiki for this play. I'll explain why I liked it below, but first I think the issue is a little more complicated than simple changes that come with the times.
To be sure, literature that represented women as "things to be submitted" was quite common in Renaissance. But this is Elisabethan England, not Salic Law France or the Iberian Peninsula. Shakespeare wrote the play when Elizabeth I had just warded off the invasion of the Armada (helped by a storm). And even in Spain, Cervantes was putting assertive women as secondary characters in Don Quijote, like the lady and her husband who fool Sancho Panza into thinking he became the king of the island Don Quijote had promised him.
My main problem with the
Shrew is that for the rest of his career, Shakespeare had many of his most fascinating, clever characters be women. And that as early as Joan of Arc in Henry VI part I (a villainness in his Anglocentric view, of course). Rosalind chooses his man and manipulates events to obtain him; Juliet tells Romeo "well, if you love me, appoint the place and time of our marriage"; Lady Macbeth is arguably the mastermind of the assassination; the Queen of France stays true to her vow of not marrying until her suitor ends his year of celibacy; Portia defeats Shylock using her cunning; Helena basically corners Bertram and leaves him no option other than marrying her, using even the King as her instrument. Mrs Page and Mrs Ford wrap Falstaff around their fingers. And so on and so forth. His weak women - Ophelia, Desdemona, Lavinia - serve very specific, tragic roles.
According to the Wiki, there's indication that the play may have caused unease in its time, with one of Shakespeare's fellow playwrights composing a sequel in which Petruchio is tamed by a second wife. Others argue that this kind of depiction of women, if not satyrical, was already behind the times by then.
I don't really have any specific knowledge of the attitudes of late 16th century England, but I feel rather comfortable with the satyrical hypothesis. I admit, though, that I ought to have written "arguably" or "possibly" instead of "probably" in my first post.
P.S.: Arguably, Petruchio does threaten to kill Kat in the play - by starvation. It's been years since I read it, but I think that's what finally breaks her. I particularly liked a BBC adaptation in which his final speech is counterbalanced by her saying that just as she is his, he is hers.
P.P.S.: I'm reminded that even
The Arabian Nights, written 700 years earlier, has a combination both of women who are completely submissive to their men and powerful women like the princess who defeats the genius in the coolest sorcery duel I've ever read in literature.