That's completely false.
The principle of mutual submission is established in Ephesians 5:21, and then the verses that follow (a modified version of the Roman 'household code') show examples of how that is done. In the Greek, there is no verb for submit in Ephesians 5:22. The verb meaning "to submit" is found only in the preceding verse (v. 21).
Let's take that 1 verse and compare it to the entirety of scriptural teaching about men ruling their households and families:
"Let the deacons be the husbands of one wife,
ruling their children and their own houses well."
1 Timothy 3:12
"For the
husband is the head of the wife even as Christ is the head of the church, his body, and is himself its Savior."
Ephesians 5:23
"
Let a woman learn quietly with all submissiveness.
I do not permit a woman to teach or to exercise authority over a man; rather, she is to remain quiet. For Adam was formed first, then Eve; and Adam was not deceived, but the woman was deceived and became a transgressor. Yet she will be saved through childbearing—if they continue in faith and love and holiness, with self-control."
1 Timothy 2:11-15
"He must
manage his own household well, with all dignity
keeping his children submissive"
1 Timothy 3:4
"But I want you to understand that the head of every man is Christ, t
he head of a wife is her husband, and the head of Christ is God."
1 Corinthians 11:3
"Likewise,
wives, be subject to your own husbands, so that even if some do not obey the word, they may be won without a word by the conduct of their wives"
1 Peter 3:1
So, either that 1 verse doesn't mean what you think it means OR that 1 verse stands in juxtaposition with the entirety of scriptural teachings. Perhaps a better interpretation of Ephesians 5:22 would be that husbands need to treat their wives with the knowledge of their weakness and therefore need not throw their weight around in non-important areas?