As the Cup of Messiah's Sacrificial Suffering is inseparable from Redemption, but Redemption is inseparable from Incarnation, so the Eucharist is inseparable from the priesthood.
The priesthood is G-d's lawful institution in Torah (not to speak about Melchizedek / Malkitsedek still before the Sinaitic Law). It is aimed to perform actions with the sense of a Sacramental Sign, e.g. showing a man that forgiven is his sin for which he had made an offering. In this way such a Sign has sequels.
Sacramental Signs witness a real (not symbolic!) mystic change associated with action for which G-d has specialised a correspondent performer, not any accidental person. All men of Levi's tribe were appointed to serve G-d, however only those of them were kohens, who were descendants of Aharon. Only kohens were capable to perform sacramental sacrifices in the Temple.
The Blessed Sacrament cannot but presume real sequels (including negative sequels – cf. the admonition of
1 Kor=Qōrintijjīm ālep 11, 29) and changes implicating G-d's gifts to one who really unites with Crucified JHS with all his heart. Since this shows a degree of perfection of the Sinaitic Covenant, similar perfection is expected for priesthood too. This seems to have been understood in apostolic times: the apostlehood was transmitted through laying on of hands. 
In the New Covenant there is a the concept of total priesthood of all believers, what has been especially developed by Protestants. Nevertheless one should not forgot that not all Levites offered sacrifices, however the kohens only. The latter correspond to bishofs in Christian tradition. The bishofs lay hands on priests consecrating them bishofs, the bishofs also ordain priests to fulfil duties. Even the reformation of Luther-Zwingli-Calvin did not change this praxis. Therefore all inventions of Neo-Protestants of the 19
th c. and later
can hardly be taken seriously. Thus all believers in JHS may be Levites, however it is a question whether all of them can be kohens!
Messianic Jews, as heirs of the whole Jewish tradition, can hardly negate priesthood of Levites and kohens of the Sinaitic Law similarly as the Sinaitic Law does not negate the priesthood of Malkitsedek. For the same reason the apostolic priesthood through laying on of the hands cannot be ignored too. 
Therefore, keeping to Jewish millenary tradition of the priesthood seems to be ideal, especially in the matter of indubitable sacraments (those with sequels) of which Eucharist is most significant according to almost all opinions.
In the future, it seems possible that the priesthgood of Malkitsedek will reappear in the renewed Church of St. Ya‘aqob with the help of messianic genetic kohens laying hands on elected priests of the priesthood of Malkitsedek. Messianic Jews are not any traditional protestants who historically come from the same Anti-Jewish Constantin's Church of 325, as Roman Catholic or Orthodoxes (cf. the beginner of Protestantism Martin Luther who was the first to transform political-religious Anti-Judaism of the Romans into future racist Anti-Semitism of the Germans). 
Although the Eucharist uniting with the Crucified JHS can be individual and spiritual, the Sign of G-d becomes apparently needed for rallying the community when delivered through the hands of G-d-elected priest.
Objections that the sign itself does not make a person participant of the Golgotos Sacrifice does not make sense, because on the one hand the Sign is given to already a participant in faith, on the other hand the Sign is Divine and mystical rather than human and material. Together with the Sign one obtains a Gift uniting all miserable or larger individual sacrifices into the Sacrifice of Golgotha. One who contributes less to the Bank of the Universal Sacrifice in Body of Saviour, that gets from there as much reinforcing Grace as it is needed for further individual steps on the way of the Lord. 
Such a kind of reasoning is certainly Catholic, however why it should be worse than any Protestant one, often rationalized and cleansed of mysticism? 

A Catholic sees in Hostia all the Body of JHS, i.e. the whole Church beginning from St. Miryam, apostles and martyres up to today living believers, ending with his neighbour who stands by his side at a Mess, who is in the state of Grace, and with whom the Community of Saints is witnessed in the same Mystical Body.
Uniting all believers round the Jewish roots, Messianic Judaism is not necessarily obliged to reject any positive experience of 2000 years of non-Jewish Christianity. Nevertheless in certain frames only. An enormous charismatic share has already brought such things into the Messianic Movement which are unthinkable in the Jewish tradition (as e.g. speaking "languages"). This can even choke up the Jewish tradition in case such believers of Non-Jewish origin prevail. 
Where no sacramental sacrifice is offered, no priesthood is necessary. Therefore it does not exist in traditioal rabbinic Judaism similarly as in Islam. Many Neo-Protestants have rejected sacramental priesthood coming from times of the Temple. This happened because of the rejection of the Papacy and the rationalisation of faith depriving it from its mystical content. However survival of pre-Nicene terminology (as bishops, presbyters) in almost all Protestant denominations reveals their Roman provenance. Therefore when confronted with Messianic Judaism, both Catholics with Orthodoxes and their opponents Protestants appear to be at the same alternative side, i.e. at a Roman side (no matter whether of "the first", or "the second" Rome). The last thing to end all these abstract speculations on "the theory of the Messianism", is a degree to which Paulinistic exegesis can be justified for messianic believers, when only one presumably Paul's epistle is addressed to the Jews, all the others are oriented toward traditions and mentality of the pagans.