Yeshua will accomplish the remaining messianic prophecies at his second coming.
What is important is having our sins forgiven through the ultimate atoning sacrifice of Yeshua and being redeemed from death. Keeping the commandments is the fruit of Yeshua's saving work.
The Kingdom of Heaven is not a magical place, but Yahweh's rule over all through His Son Yeshua. As Daniel 2:44-45 reveals, the Kingdom of Heaven will come down (with Yeshua as its King) and destroy all earthly kingdoms. This Kingdom will rule the earth for 1,000 years until sin and death are totally destroyed.
My parents told me my ancestors were Italian. My ancestors probably did the same with their children and so on. However, somewhere along the line, the history of how my ancestors got to Italy in the first place was lost. Perhaps they migrated from Israel and I'm really Jewish. The point is, how things transpired over thousands of years can easily get distorted. Hopefully your ancestry traces back to the patriarch Israel. However, if your far distant ancestors were really Edomites, for example, they could have lied about being Jewish to have a better life among other Israelites.
Look at the example of what happened in Ezra 2:61-62. These people claimed to be of priestly lineage, but their names were not found in the registry. So they were kept out of the priesthood. How much more could such things happen after the registries were lost. Had there been no registry at that time, those people would have been accepted as priests.