You are wrong.
There are two clauses that comprise the Second Amendment, an operative clause, and a prefatory clause.
Operative clause: "The right of the people to keep and bear Arms, shall not be infringed."
The operative clause is the actual protected right. "Right of the People" is used 3 times in Bill of Rights. All three of these instances unambiguously refer to individual rights, not "collective" rights, or rights that may be exercised only through participation in some corporate body.
Prefatory clause: "A well regulated Militia, being necessary to the security of a free State."
The prefatory clause is the lead-in that announces a purpose for the operative clause, not the purpose. It does not limit or expand the scope of the second part, the operative clause. The operative clause’s text and history demonstrate that it connotes an individual right to keep and bear arms. All of the period material surrounding the debate of the Second Amendment shows that the government is the restricted party, not the People. "Free State" refers to the state of being free, not one of the 13 colonies.
The Amendment could be rephrased, "Because a well regulated Militia is necessary to remain free from government oppression, the right of the people to keep and bear Arms shall not be infringed."