Of course you will find texts based on Scripture which say, as Scripture does, that the spiritual head of the family is the husband - and that this leadership is enacted in Christ-like, self-giving sacrificial love.
What you are not going to find is a text saying that women are perpetual minors who always have to be under some male’s authority, because that doesn’t exist.
“Giving away” one’s daughter is a cultural custom linked to the times when marriages were, in part at least, a financial transaction between two families. It has nothing whatsoever to do with the sacrament of matrimony and its meaning.
Where I live, the custom for weddings is that the groom processes in escorted by his mother, and the bride processes in escorted by her father. It’s just a way of including the parents, and if there was to be a symbolism in it, it would be at most that the old family units are shifted to form a new one - that “man leaves his father and mother and is united to his wife, and they become one flesh” (Gen 2:24).
The sacramental bond which creates the relationship you’re talking of between a man and a woman is precisely between a man and a woman, not between a man and a future, not yet extent, hypothetical family.
From the CCC:
2217 As long as a child lives at home with his parents, the child should obey his parents in all that they ask of him when it is for his good or that of the family.
2232 Family ties are important but not absolute. Just as the child grows to maturity and human and spiritual autonomy, so his unique vocation which comes from God asserts itself more clearly and forcefully.
By that logic, fathers should give away their sons as well.