I don't know about every one of those, but "Jew" is more than the religion. Being Jewish is a...culture and a people. One is born a Jew if one's mother is, no matter what one's beliefs are. I think that many Messianic Jews are simply acknowledging that; they want to be a part of that culture and people, with the Messiah already come.
That is their belief and I see nothing wrong with it. I especially see nothing wrong with wanting to be a part of a group that has been around, and culturally strong, for thousands of years. Very few, if any, other human group, has done that. I am not Jewish in any way, but if I had Jewish ancestry, I would want to be a part of that culture and group.
But I wouldn't change my own religious beliefs to do so.
Religious beliefs are personal, and it doesn't really matter what anybody else thinks, does it?
You are certainly enveloping to define what the word "Jew" means to you, as fed to you no doubt by the matrilineal bloodlines which many understand to be racist & cultic, who have declared a monopoly over "right & wrong, property, inheritance, law, & God by genetic heritage".
Jesus never came to eradicate the law, either; so the term "Christian" is a modern perversion of the term "Jew", because up until recently where slang has lazily began dropping the word "Judeo" from Judeo-Christianity, Judeo-Christianity has always and literally truly meant by 100% in all fulfillments of Old Testament promises to literally be "Jews who believe Christ is the messiah".
As with all idiocy in mankind, language becomes hijacked, divided, and people cling to the linguistics of their mass media, culture, school corporate funded textbooks, and childhood ears. But at no point has Judeo-Christianity meant anything other than another denomination of Jew.
The whole concept of "Messianic Jews" is redundant. So to complete my point here, your reiteration that being a Jew is only true by matrilineal denominations, is feeding right back into hateful regime of "religious wars" where everyone is supposedly divided, and nobody is allowed to have access to God except those in some kind of monarchial "royal gene pool".
The same matrilineal supposed "one and only true Jews who are destined to possess the world itself by God's divine genetic favoritism", have indeed convinced the Judeo-Christians that "Christianity" & Judaism are & always have been separate"—subliminally asserting their disbelief cosmically that He was not the messiah.
But let it never be mistook—that Judeo-Christianity refers to a denomination of people also identifying as "the one true" Jews. It is by capitalist perversion and devolution that these sects have grown to forget that Jesus announced Himself the messiah of the Jews, which includes His believers. There is no "Christian" vs. "Jew", there is and always has only been "Jew".
Judeo-Christianity is just another denomination of Judaism, and always has been; "Messianic Jew", again, is redundant.