Do messianic Jews claim to be the beneficiaries of the new covenant now?

We are currently apart of the new covenant.

The new covenant reaches further fulfillment on Jesus' return - then "the knowledge of the Lord will be as the waters cover the sea."

Currently, we are in a transition phase as we await the fulfillment of Ezekiel 37-38 and other end times prophecies.

Galatians 4:20-31

20-21 -I stand in doubt of you. Tell me, ye that desire to be under the law, do ye not hear the law?

22 For it is written, that Abraham had two sons, the one by a bondmaid, the other by a freewoman. 23-24 BUT HE WHO WAS OF THE BONDWOMAN WAS BORN AFTER THE FLESH; but he of the freewoman was by promise. Which things are an allegory (allégoreó): for these are the 2 covenants; the one from the mount Sinai, which gendereth to bondage, which is Agar.

25 For this Agar is mount Sinai in Arabia, and answereth to Jerusalem which now is, and is in bondage with her children.

26 But Jerusalem which is above is free, which is the mother of us all.

27 For it is written, Rejoice, thou barren that bearest not; break forth and cry, thou that travailest not: for the desolate hath many more children than she which hath an husband.

28 Now we, brethren, as Isaac was, are the children of promise.

29 But as then he that was born after the flesh persecuted him that was born after the Spirit, even so it is now. Nevertheless what say the scripture? 30 Cast out the bondwoman and her son: FOR THE SON of the bondwoman shall not be heir with the son of the freewoman.

31 So then, brethren, we are not children of the bondwoman, but of the free.

V28-29: Notice there were already those born after the spirit in Abraham’s time.

• (Bondswoman) Agar is figuratively likened to the Old Covenant (Mount Sinai) and represents individuals of her whom are bound to sin and born by the flesh.
• (Freewoman) Sarah is figuratively likened to the New Covenant (New Jerusalem) and represents individuals of her whom are free from sin and born by the spirit.

In Genesis, God promised Abraham a son through his wife Sarai (Sarah) whom was infertile (Ge. 15:2-4, 16:1-2, 18:12-15). This son was Isaac, who was born because of a divine promise from God and God made his covenant with him (Ge. 17:15-21) and the 12 tribe tribes of Israel descended from him. Abraham and Sarah decided for Abraham to have another son with his other wife who was a bondwoman (Ge. 16:15). This son he had by Agar was Ishmael, who did not come about due to a divine promise but a natural fleshly way. Ishmael also mocked Isaac, and was cast out of Abraham’s house which God approved of (Ge. 21:8-12).

It should be made clear that the meaning of this story is at the forefront: that God will toss out the bondwoman’s son (those born after the flesh: v23, 30). Ultimately, this allegory boils down to being a child of promise accepted by God for New Jerusalem, while the story’s persons, then these person’s traits which are assigned to the covenants are used together to illustrate this hidden lesson. This is not about God’s promises made to the national body of Israel (Ro. 9:3-4), and the assigned traits are not all inclusive for every individual. David, Moses and the other prophets from the Old Covenant will still be ‘children of the free women,’ (Lu. 13:28-29) and so will Abraham because he had faith in God’s word despite not being under either covenant. This allegory seems to have been given as a warning to those Galatians seen in the previous chapter (Gal. 3:1-5) who had gone back to walking by the flesh (being under the law of sin).

Messianic Passages: [A]

Passages regarding Israel, the prophetic timeline, and the covenants: [1] [2]

Source: Self - Messianic Jew: 3,000+ hours of Bible study.

This will probably answer alot of your questions:

http://www.messianic.org.nz/circumcision.pdf

One thing to really be aware of, is like Protestants, Catholics and others - those who call themselves "Messianic Jews" can have very different views on scripture. But this will give you the view of this group of Messianic Jews, how they view the new covenant, and how Gentiles fit in to it.

I knew a Messianic Rabbi years ago, he and his congregation definitely accepted the New Covenant and the same goes for others I have since met online. (Meaning they accepted Christ as Savior, accepted the Council of the Book of Acts that gentiles are not bound by the Mosaic law etc.).

Anyway, that rabbi tended to relate Messianic Judaism along the lines of "Wild" and "Natural" Olive Tree branches quoted at the end of this post. His ambition was to be an evangelistic outreach to traditional Jews, as well as provide a spiritual home for them, and he wanted to be a resource for gentiles that would show them the Jewish context of the Gospel and the rest of the Bible (I think he wanted to do a lot the stuff that Jonathan Cahn currently does).

Romans 11
13 I am talking to you Gentiles. Inasmuch as I am the apostle to the Gentiles, I take pride in my ministry 14 in the hope that I may somehow arouse my own people to envy and save some of them. 15 For if their rejection brought reconciliation to the world, what will their acceptance be but life from the dead? 16 If the part of the dough offered as firstfruits is holy, then the whole batch is holy; if the root is holy, so are the branches.

17 If some of the branches have been broken off, and you, though a wild olive shoot, have been grafted in among the others and now share in the nourishing sap from the olive root, 18 do not consider yourself to be superior to those other branches. If you do, consider this: You do not support the root, but the root supports you. 19 You will say then, “Branches were broken off so that I could be grafted in.” 20 Granted. But they were broken off because of unbelief, and you stand by faith. Do not be arrogant, but tremble. 21 For if God did not spare the natural branches, he will not spare you either.