But couldn’t a Gentile get there by converting to Judaism?
Your post got me to thinking that one problem Christians have today is rejecting the Jewishness of Jesus. Do Christians observe the Lev 23 Feast days? Do they celebrate Easter and Christmas, or follow the Bible days of First Fruits and Tabernacles? When Constantine took over God's church, he was antisemitic and changed times and dates and laws away from the Bible. Do you understand the Sabbath is the 7th day or the 1st day? Jesus didn't change anything but He came to fulfill everything. Jesus didn't come to change the Law but to fulfill it.
By understanding Scriptures, we can understand much more than what a Gentile Christian is exposed to. Ask the average Christian what the "Day no one knows" is and they will say you can't know. A Jew will tell you it's Rosh Hashanah or Trumpets. A Messianic Jew just looks at Scripture and see how Jesus fulfilled all the Law for us.
So is a Messianic Jew a Christian or a Jew? The name "Christian" was used as a bad name for followers of Christ, but they are "believers" in Scripture.