Yes, they would be considered Messianic Jews.....
No, they would not. If that is their motivation, they have been profoundly poorly taught.
Paul's letter to the Galatian church would be helpful to review.
Look: I, Paul, say to you that if you accept circumcision, Christ will be of no advantage to you.
3 I testify again to every man who accepts circumcision that he is obligated to keep the whole law.
4 You are severed from Christ, you who would be justified by the law; you have fallen away from grace.
Well then what was Jesus’s Mother? what about Mary of Magdela? What about the Jews in Corinth?
What about the Jews who did not accept Pauls’ or Constatines teachings?
The Jews who knew and accepted Jesus in Israel as the Chosen, the Messiah still adhered to Jewish Laws and Customs.