What was the purpose of Jesus death for the Jews?

During the last supper that Jesus had with His disciples, He said the following to them,

Luke 22:20 Likewise He also took the cup after supper, saying, this cup is the new covenant in My blood, which is shed for you.

One must remember that since the time of Moses and the exodus, the Jews have been under a covenant of Law, the 'old' covenant. This covenant was a burden (Matthew 11:28; Acts 15:10) and would never be able to be fulfilled by the Jews. Its primary goal was to teach them that they needed a Saviour, and indeed, to point them to their Messiah (Galatians 3:24-25). The 'old' covenant needed to be replaced, and the first advent of Jesus was exactly for this purpose, as is confirmed in the above verse. Now, if you focus on the bolded part you will see that it alludes to HOW Jesus was going to do it.

In order to replace the old covenant with the new covenant, Jesus had to die. We are familiar with the legalities of our current day 'last will and testament'. In general terms, a child can only inherit the estate of their parent(s) when the parent(s) die. The will and testament cannot become active until the testator(s) have passed away. Well, this is exactly the reason why Jesus had to die for the Jews. You see, Jesus was the testator of a new 'will and testament'. The book of Hebrews confirms this fact accurately,

Hebrews 9:13 For if the blood of bulls and of goats, and the ashes of an heifer sprinkling the unclean, sanctifieth to the purifying of the flesh: 14 How much more shall the blood of Christ, who through the eternal Spirit offered himself without spot to God, purge your conscience from dead works to serve the living God? 15 And for this cause he is the mediator of the new testament, that by means of death, for the redemption of the transgressions that were under the first testament, they which are called might receive the promise of eternal inheritance. 16 For where a testament is, there must also of necessity be the death of the testator. 17 For a testament is of force after men are dead: otherwise it is of no strength at all while the testator liveth.

If the purpose of Jesus death was to ratify a new covenant to replace the old law covenant, then what was the new covenant? The purpose of the new covenant was to set up the Jews to receive their inheritance. The law could never save, the law could never provide forgiveness, the law could never be realized or achieved by sinful men. Through the death of Christ, the testator, the new covenant would bring about forgiveness (taking away) of sin and the indwelling Holy Spirit who would write a higher and more perfect law on their heart and enable, or empower them to know it and live it out naturally, (not from the flesh, or through effort, or from memory). This new covenant would be in fulfillment of the prophecies we read in Jeremiah and Ezekiel,

Jeremiah 31:31 Behold, the days are coming, says the Lord, when I will make a new covenant with the house of Israel and with the house of Judah— 32 not according to the covenant that I made with their fathers in the day that I took them by the hand to lead them out of the land of Egypt, My covenant which they broke, though I was a husband to them, says the Lord. 33 But this is the covenant that I will make with the house of Israel after those days, says the Lord: I will put My law in their minds, and write it on their hearts; and I will be their God, and they shall be My people. 34 No more shall every man teach his neighbour, and every man his brother, saying, 'Know the Lord,' for they all shall know Me, from the least of them to the greatest of them, says the Lord. For I will forgive their iniquity [immorality, offences], and their sin I will remember no more. (See also: Ezekiel 11:16–20, Ezekiel 36:22–30).

For the Jews, Jesus' death was more a legal matter than a sin matter. The covenant itself dealt with the matter of sin, so Jesus' death was for activating the covenant that would bring salvation to the Jews. As for the Gentiles, they did not have a covenant that legally dealt with sin and salvation, so Jesus' death was for a whole other purpose; it was a payment for redemption, it was propitiatory, it was taking our place in order to exchange sin for righteousness and wrath for grace. It was the ultimate sacrifice.