I agree wholeheartedly. Neither has the Executive or any other branch of Government. The Constitution trumps any treaty, if we follow the Constitution. Unfortunately, there seem still to be differences over what the shall not be infringed part of the Second Amendment actually means. It has been infringed frequently.
The Constitution does not include in the enumerated rights set forth in the Bill of Rights or elsewhere such things as a right to drive an automobile, to practice law or medicine, etc. Those have been treated as privileges rather than as rights. The exercise of privileges perhaps should require State or Federal licenses. The exercise of rights, such as the right to bear arms, should not.
I don't understand the constitutional bases for requiring permits -- similar to but in some cases more stringent than drivers' licenses -- to own and to use firearms,. Yet we do that.
Should the new arms control treaty be ratified by the Senate (on a 2/3 vote, as required under Article II), Federal implementation seems likely to create another morass of Federal gun control laws. Should it? Of course not. Will it? That's a different topic altogether. I suspect that it is likely to do so.
Even should the Senate not ratify the UN treaty, there is a possibility that the President, by Executive decree Order might try to implement it domestically. What will we do then? Cry? I don't know and hope not to find out.
In real life, treaty law trumps the constitution.