Notice here that Jesus does not say after this and other examples you quote, "but from the beginning it was not so. And I say unto you..." Pay attention to the details.
Matthew 19
4 And he answered and said unto them, Have ye not read, that he which made them at the beginning made them male and female,
5 And said, For this cause shall a man leave father and mother, and shall cleave to his wife: and they twain shall be one flesh?
6 Wherefore they are no more twain, but one flesh. What therefore God hath joined together, let not man put asunder.
7 They say unto him, Why did Moses then
command to give a writing of divorcement, and to 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.
Moses was a human being. He made judgments and decisions. As you pointed out, the scribes and Pharisees say in Moses' seat, and while Christ's disciples were to honor that, were their decisions all inspired by God? Could they make a court decision that was wrong? Was Hillel right in saying a man could divorce his wife if she burnt the food? Were they right about oaths? And if the decisions of Jewish authorities directly contradicted obedience, to God, the apostles decided that they had to obey God rather than men. The Saducees did not want the apostles preaching the doctrine of the resurrection, and the resurrection of Christ, in the temple, but they had to obey God rather than men. Peter realized that he was supposed to go under a Gentile roof.
Why would whatever decisions Moses had made or allowed to occur that set up the case described in Deuteronomy 24 have to be inspired by God? Deuteronomy 24 is a case.
Look at the wording of the passage in question, Deuteronomy 24. We should interpret it consistent with how the Jesus did, not how the Pharisees did.
Deuteronomy
24 “When a man takes a wife and marries her, and it happens that she finds no favor in his eyes because he has found some uncleanness in her, and he writes her a certificate of divorce, puts it in her hand, and sends her out of his house, 2 when she has departed from his house, and goes and becomes another man’s wife, 3 if the latter husband detests her and writes her a certificate of divorce, puts it in her hand, and sends her out of his house, or if the latter husband dies who took her as his wife, 4 then her former husband who divorced her must not take her back to be his wife after she has been defiled; for that is an abomination before the Lord, and you shall not bring sin on the land which the Lord your God is giving you as an inheritance. (NKJV)
The kind of stuff that people might do with their marriages is found in verses 1 through 3, but the command is given in verse 4. In this situation where a man would give a certificate to his wife and the wife marry someone else, and he dies or divorces her... the former husband must not take her back.
How do we know that is the interpretation? From the Lord Jesus. He said that Moses because of the hardness of your hearts allowed divorce, but from the beginning it was not so. Do not mistake this for Jesus disagreeing with scripture here in Deuteronomy 24 ( @NOV25 @SomeDisciple ). This passage is setting up a case for how divorces worked among them. We know from Christ that this divorce certificate stuff was allowed by Moses because he said so.
Like the Pharisees, you are treating the divorce certificate business as something commanded by God, when Christ is treating it as something permitted by Moses.
Take a look at the wording of the pasage in Deuteronomy 24 in Hebrew. Notice it starts with 'ki', setting up a condition 'when' or 'if' scenario... laying out a case.
https://biblehub.com/interlinear/deuteronomy/24.htm
Now, compare that to this scenario in the next chapter, 25:11-12.
https://biblehub.com/interlinear/deuteronomy/25.htm
11 When men strive together one with another, and the wife of the one draweth near for to deliver her husband out of the hand of him that smiteth him, and putteth forth her hand, and taketh him by the secrets:
12 Then thou shalt cut off her hand, thine eye shall not pity her.
(KJV)
If you interpreted this passage the same way you and the Pharisees and @NOV25 interpreted the divorce certificate scenario in Dueteronomy 24, then you would have to interpret this passage to mean
When two men strive together with one another, and the wife... SHE IS COMMANDED TO TAKE HIM BY THE SECRETS, and then they are commanded to cut off her hand. But it doesn't say that.
What if you were to interpret this verse the same way you interpret the divorce passage in Deuteronomy 24?
Deuteronomy 22
23 If a damsel that is a virgin be betrothed unto an husband, and a man find her in the city, and lie with her;
So that would become "... if a man find her in the city...HE SHALL LIE WITH HER." If you interpret the same way you interpret Deuteronomy 24, then this verse is commanding men to fornicate with or rape other men's betrothed. Clearly that is that what the passage is saying.
https://biblehub.com/interlinear/deuteronomy/22.htm
We must interpret Deuteronomy 24 consistently with the way Moses does.
So Moses gave them what God gave to Moses. Nothing originated with Moses.
I don't think you really believe this. Would you blame God for Moses trying to get out what God was telling him to do in the burning bush passage? Would you blame God for Moses killing the Egyptian? What about not circumcising Gershom, or striking the rock twice?
Therefore it is God who permitted divorce in the Torah, even though Christ explained that it was only because of the hardness of the hearts of men.
You need to read all the words in a passage to understand what it is saying, not just skim it.