The wives put away
with a divorce certificate were 'put away' wives also. in fact, in Matthew 19, the topic the Pharisees were asking Jesus about was about when one could 'legally' put away a wife--- with a certificate.
7 They say unto him, Why did Moses then command
to give a writing of divorcement, and [quoto put her away?
8 He saith unto them, Moses because of the hardness of your hearts suffered you to put away your wives: but from the beginning it was not so.
9 And I say unto you, Whosoever shall put away his wife, except it be for fornication, and shall marry another, committeth adultery: and whoso marrieth her which is put away doth commit adultery.
Look at the topic they brought up,
3 The Pharisees also came unto him, tempting him, and saying unto him, Is it lawful for a man to put away his wife
for every cause?
You can find articles on the Pharisaical debate on the 'any cause' divorce. In the generation before Christ and around the time of his birth, the 'nasi' or prince of the Sanhedrin, the most prominent scholar was Hillel. The second most influential leader was Shammai. Their students disagree with each other. The Jerusalem Talmud even says some of the Shammai school killed off some of the Hillel school. I hear this was at a Bible study.
But they also debated about a number of things. Hillel was more friendly toward Gentile conversion to Judaism and made it easy. Modern Orthodox Judaism evolved out of Hillel Phariseeism. The Shammai folks were in power for much of the New Testament. The idea that going into a Gentile's house was unlawful might have been a House of Shammai decision.
When it came to divorce, Shammai interpreted Deuteronomy 24 to allow for a divorce certificate if some form of uncleanness were found in a woman-- adultery or maybe something a bit less extreme than that, but still serious. Hillel allowed for divorce 'for any cause.' if a woman burnt the bread, he thought a man could give her a divorce certificate.
So when we read about 'divorce for every cause' or for any cause in the passage, we should think of the debates referred to in their own literatura, written down later of course as a record of their 'oral law' in the Mishneh or Talmud.
Here is a quote about three falsely so-called 'rabbis'-- one of them who supported a false Messiah after Christ. The first two speaking before Matthew 19 was written. This is a quote from a Jewish site:
https://www.chabad.org/library/article_cdo/aid/2285/jewish/When-to-Get-Divorced.htm
At the time Matthew 19 happened, the first two opinions were being debated.
I am not sure if this is where you are going, but I have seen a whole website and a number of posts on Internet forums defending the idea that a Christian man can divorce his wife for any reason if he gives her a divorce certificate. That is completely contrary to Christ's teaching in Matthew 19. If all he were saying was, "You had better give her a piece of paper that says you are divorced", that was already proper doctrine the Pharisees would have agreed on, and the apostles would not have responded, "If such is the case of a man with his wife, it is better for a man not to marry." Giving a woman divorce certificate was already accepted practice.
Those who did not do so were not following the Pharisees prescriptions. They were trying to get around the law.
This trend toward interpretation Matthew 19 relies on ignorance of the historical setting and also seems to be an attempt to lower the bar of Biblical morality to conform with the rampant disobedience to the teaching of Christ and Paul on the matter that we see in so many churches today.