I'd say that Christianity is not the perfection of Judaism.
I would say that a refined Messianic Jewish faith (included in that Messianic Gentiles who are true to their own cultural heritage and are part of the commonwealth of Israel.) is a more accurate continuation of Jewish faith.
I (we?) believe Israel (specifically Jerusalem) will be the 'capitol' of the world and Messiah Yeshua will rule as King over the earth. All the Nations must pay tribute to the King and observe certain feasts and appointed times. (Found in Torah.) That the Living Torah Himself will reign. (Thus, complete fulfillment.)
When Messiah comes (again) the fullness of what Torah foreshadowed will be revealed. The first time, He came to enlighten us through Torah teaching and be the sacrificial Lamb and First Fruits of the resurrection. Next, He will come to defeat Israel's enemies and rule as King.
A large portion of the Christian community believes that Christianity (a religion separate from Judaism) will be the ruling faith and the new Israel. Jews will convert to Christianity. They will accept a new religion, life style, etc. Some support Israel only because they think a Gentile Jesus is coming and they will all be whisked away and leave the Jews to battle out Armageddon.
Well, we think Yeshua will reveal Himself to all of Israel... and many will be surprised to find a JEWISH messiah ruling in a Jewish context. The Christian world might be slightly surprised when mass 'conversions' to a different faith don't take place- instead- Jewish believers remain Jewish and walk in 'completeness' :Torah & Messiah Yeshua.
When Israel sees Yeshua as her Messiah it will be a big "Ah-ha" moment. And a time of joy & sorrw. Not because they realize "We should have changed our religion and followed the Christian god." But more like, "We should have know our own Messiah!"
Many in the MJ community believe the number of believing Jews in Israel will greatly increase right before Messiah returns. And Israel herself will call upon Yeshua to save them- which He does.