Before I start exploring though I was wondering whether its still generally quite common for Christian men to desire a woman to submit to them, or whether its now more common to want an equal partnership?
If they don't, they're either being disobedient or they haven't got up to that bit in the Bible yet.
Regardless, that call to submission is tempered with a call on a guy to love your wife as your own body. In addition, a guy has church authority over him, and if that church is functioning as it ought to (and sadly many don't), it will call him out if he's not loving his wife properly.
So in effect, while not equality in the sense of doing away with differing gender roles (as, say, feminists would understand equality), it's equality in terms of respect.
The relevant Bible passage, for reference:
Ephesians 5:22-33
Wives, submit yourselves to your own husbands as you do to the Lord. For the husband is the head of the wife as Christ is the head of the church, his body, of which he is the Savior. Now as the church submits to Christ, so also wives should submit to their husbands in everything.
Husbands, love your wives, just as Christ loved the church and gave himself up for her to make her holy, cleansing her by the washing with water through the word, and to present her to himself as a radiant church, without stain or wrinkle or any other blemish, but holy and blameless. In this same way, husbands ought to love their wives as their own bodies. He who loves his wife loves himself. After all, no one ever hated their own body, but they feed and care for their body, just as Christ does the church— for we are members of his body. “For this reason a man will leave his father and mother and be united to his wife, and the two will become one flesh.” This is a profound mystery—but I am talking about Christ and the church. However, each one of you also must love his wife as he loves himself, and the wife must respect her husband.
So if a Christian guy ever thinks he is being granted a right to dictate to his wife based on whatever stupid whim he has, he doesn't. Of course people try to spin it this way, for no good reason.
More info here:
https://www.gotquestions.org/wives-submit.html