I don't know the answer to this, but I want to propose a theory. If a man were to marry, he paid a bride price. This money, as I understand it, was no inconsequential amount, and the bride could only redeem this money in the event of a divorce or the husband's death. It could be that sending a wife away without providing a certificate of divorce was manipulating the law in order to avoid allowing the wife to redeem the bride price. After being sent away, she would be unable to provide for herself easily outside of prostitution, and she might find another man to provide for her in exchange for relations with her, which is committing adultery. Once this occurred, she was no longer entitled to redeeming the bride price.
I think you've got the bride price and redeeming a bit mixed up. The concept of "redeeming" appears throughout OT law in relation to land, livestock, and slaves, and these verses help us understand what it is about to apply it to marriage.
Redemption is simply "buying back what once belonged to you".
For instance, if a man sold a house in a city, he had the option of buying it back for a year. This avoided rash decisions that had long-term consequences for the inheritance of children. If he didn't buy it back (redeem it) within a year, it was to remain the property of the buyer forever (Leviticus 25:29-30).
The firstborn of a man's livestock belonged to YHWH. The firstborn of unclean livestock (which could not be sacrificed) could either be redeemed (purchased back from YHWH) by paying the value plus 20% to the temple, or they would be sold and the profit go to the temple (Lev 27:26-27).
If a man was sold as a slave, his family could redeem him by buying him back for the value of his labour until the next year of jubilee, or the original price he was sold for, depending on the situation (Lev 25:47-55).
In all cases, redemption always involves a transfer of money from the original owner / family of the item / person in question, to the current owner. The person being redeemed does not receive any money at all.
When Christ redeems us from our sins, we don't receive any payment. We are slaves to sin, but he pays the penalty for our sins, the price required to redeem us, so we can be set free.
This is also shown by the exceptions.
Slave-women could also be redeemed for money, just as a male slave could be (ie their family could buy them back). However, if they were taken as a wife but various obligations were not fulfilled, they would go free without payment - in other words their family could claim them back without having to pay anything. (Ex 21:7-11).
Any slave, if physically mistreated, was to be released free without payment - again they could return to their family without their family having to buy them back (Ex 21:26-27)
Now, if we bring the concept of redemption into marriage and the bride-price (I'm not convinced it is relevant, just hypothetically), what would it mean for a wife to be "redeemed"?
On marriage, the husband paid a bride-price to the father.
If the father were to redeem the woman, he would pay the bride-price back to the husband, and take his daughter back.
So if divorce involves "redemption", the person profiting financially would be the husband, NOT the wife. A man who dislikes his wife would actually WANT her to be redeemend, so he got back the money he had paid for her. So basically, I think you've got this all backwards, to be honest. It's not a valid argument.
However considering these verses, a completely different line of reasoning does arise. If a slave suffers physical harm (defined), they are to be released free without payment (Ex 21:26-27). If a wife has at least the same, if not greater, rights than a slave, then she too surely has the right to be released free from an abusive husband. I am unsure whether this would involve "putting away" or "divorce", something to ponder further.