Given that the Privileges or Immunities Clause states that it only
applies to citizens (No State shall make or enforce any law which
shall abridge the privileges or immunities of citizens of the United
States), would a right incorporated through the Privileges or
Immunities Clause apply to non-citizens?
Under this theory the Second Amendment right and any other right incorporated through the 14th Amendment privileges and immunities clause would not protect non-citizens.
This said, due to the Slaughter-House cases (1873) and some related cases (which incidentally were probably decided contrary to the intent of the drafters of the 14th Amendment by conservative judges in office at the time), the privileges and immunities clause of the 14th Amendment is almost a nullity since it has been held to apply to only privileges and immunities that are specifically contingent upon United States citizenship (e.g. the right to enter the U.S. from abroad, and the right to vote in federal elections).
Since the process of incorporating the Bill of Rights to apply against the states began historically after the Slaughter-House cases were decided, the U.S. Supreme Court instead turned to the due process clause of the 14th Amendment to make most provisions of the Bill of Rights applicable to U.S. states.
Justice Thomas is an extreme outlier in judicial opinion regarding the notion of incorporating any of the provisions of the Bill of Rights to apply to the states via the privileges and immunities clause of the 14th Amendment. No other justices on the U.S. Supreme Court and very few legal scholars (and none who are particular notable) support this legal theory.
Liberal legal scholars have been reluctant to suggest incorporation through the 14th Amendment privileges and immunities clause because they are aware that it would only apply to U.S. citizens, and also because they think there is merit in the underlying concept of substantive due process behind incorporating the Bill of Rights (selectively) to apply to state and local governments through the 14th Amendment due process clause (which also provides a theory that makes sense concerning which Bill of Rights protections should and should not be incorporated).
Conservative legal scholars have more often been skeptical of the entire concept of incorporating the Bill of Rights to apply against state and local government, even though this is a well settled fait accompli because there are provisions such as the establishment clause of the 1st Amendment and the 4th and 6th Amendment exclusionary rules, which they disagree should be universally applied.
The reason that Justice Thomas is interested in this alternative route to incorporation is that he disfavors the concept of "substantive due process" that was behind the Roe v. Wade decision and a variety of unenumerated rights (e.g. the right to contraception) that arise from a substantive due process analysis. But, one can't entirely abandon substantive due process as a legal doctrine without also making all of the provisions of the Bill of Rights inapplicable to state and local governments, which is a bridge too far even for an extremely conservative justice like Justice Thomas.