Why is Messianic "Judaism" so popular among so many denominations of Christianity? The most obvious answer is because Messianic "Judaism" is a creation, and a branch of, Christianity designed to convert Jews. That is true but most Christians do not know that Messianic "Judaism" was created by the Church of England in the 1800's so this does not answer the question fully. Another answer is because so many different Christian denominations sponsor Messianic "Jewish" ministries and congregations. Again, this is true but it does not answer why these denominations would have decided on this sponsorship to begin with. A further possibility is because Paul said that Christianity should be spread "to the Jew first and then to the Romans." This claim certainly motivates Christians but it does not answer why Paul would have wanted the Jews "first." It lacks the nuance of the psychology behind this desire and why this desire is so strong, still 2000 years later while other early Christian concepts have disappeared. Further, it does not explain why post-conversion to Christianity, many of these new Christians are still identified with Judaism (e.g. former British Prime Minister Benjamin Disraeli, whose family converted when he was a child.) One more possibility is because Christians want to get back to their "Jewish roots." This response is also valid despite the fact that Messianic "Jewish" practice does not look similar to the likely practice that Jesus and his followers would have engaged in on a daily basis. This response highlights the general ignorance of the majority of Christians to the Jewish Bible and Jewish practice. Further, it does not answer the question as to why these "Jewish roots" are even valuable to Christians when Christianity has so many different beliefs from Judaism.
To understand the real psychological reason behind this popularity, we must look at the conflicted historical relationship that Christianity has with Judaism. This is probably best illuminated in the famous "Jewish Question." The "Jewish Question" was based in the following conundrum; on one hand, Christians cannot fully denigrate Judaism since Christianity claims to come from Judaism and be based in the Jewish Bible. If Christianity attacks Judaism too roughly, Christianity cuts out what it believes to be the basis of its own faith. On the other hand, the very presence of Jews stands out as a clear rebuttal that the Christian view is wrong. After all, if Christianity is correct, why does Judaism still exist in its midst? This "Jewish Question" has motivated countless debates, enactments and actions throughout the course of history. For example, in the early 1900's the Russian government mulled over a 1/3 policy: kill 1/3 of their Jewish population, push out 1/3 of their Jewish population out of their country and force 1/3 of their Jewish population to convert. Though this policy never came to fruition, it certainly demonstrates the depth of the conflict in the hearts and minds of Christians.
So then, of what value is the conversion of Jews beyond that of any other convert to Christianity? When a non-Jew converts to Christianity it is merely one more member to join the Christian "flock." However, when a Jew converts to Christianity, many Christians view it as a legitimization of Christian belief. Since Christianity came out of Judaism, a Jewish convert is viewed as a validation of Christianity's split from Judaism. For instance, this attempt for validation is seen in Reformation and post-Reformation era desires in England to bring Jews back to England following centuries of expulsion. If the English could bring the Jews back and then convince them to join Christianity, then the English churches could see this as a proof of their religious correctness.
This brings up an obvious question, namely why the use of Messianic "Judaism" to convert Jews appears to be so much more loved by Christians than other missionary methods? I believe that the reason is because the "Jewishness" in the presentation of Messianic "Judaism" is a present rather than a past activity of the convert. In more typical methods of conversion, once the Jew converts and is assimilated into the greater Christian congregation, the perceived validation received by the conversion fades quickly. For a brief period of time the Christians may feel validated in their belief by having a Jew convert but what is there to remind the Christian that this person was once a Jew after their conversion? The convert looks, acts and behaves like a Christian. With Messianic "Judaism", the Jewish-style practice serves as a regular reminder of the convert's Jewish past, and subsequently it serves as a more consistent perceived justification of the Christian's belief system.
Throughout history there is an abundance of Christian writings on the importance of converting Jews as proof of the correctness of Christian belief. I believe that the method of Messianic "Jewish" presentation functions to causes Christians to emotionally receive this "ego boost" of belief more often than other methods of conversion and this is the underlying psychological reason to Messianic "Judaism" popularity in Christianity over other methods of Jewish conversion.