In any case Jesus never broke one part of the law nor taught anyone to break it. He wouldn't advocate divorce for something the law commanded death for. He wouldn't have brought it up as a Jew talking to Jews under law in a discussion about divorce.
This is an old debate, of course, the 'traditional view' versus the view that many Protestant's held to. There are those who would argue that the Greek rendered 'except it be for fornication' means setting aside the issue of fornication, not addressing it at the time.
The situation Jesus referred to when he wrote, 'except it be for fornication' was a man divorcing his wife, though, not a wife divorcing her husband. Wives divorcing their husbands was not a legal thing in the Old Testament. Polygamy was also practiced in the Old Testament, but polyandry was not allowed. IMO, polygyny is inconsistent with what Christ revealed in Matthew 19 also. But we should keep the polygamy situation in mind when considering that males and females did not have recriprocal roles and obligations in marriage in the Old Testament.
The teachings in Mark and Luke do not have the exception clause about adultery.
Mark 10
10 In the house His disciples also asked Him again about the same
matter. 11 So He said to them, “Whoever divorces his wife and marries another commits adultery against her. 12 And if a woman divorces her husband and marries another, she commits adultery.”
Luke 16:18
18 “Whoever divorces his wife and marries another commits adultery; and whoever marries her who is divorced from
her husband commits adultery.
Here is the verse from Matthew 19 again
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.
The 'whoever's and 'whoso' in Mark and Matthew were enough to remove divorced candidates from my consideration set.
Nowadays, both men and women can work in most of our societies if they get a divorce. Celibacy is an economically viable option.