You are speaking of the Convention of the States option of formulating amendment proposals, not the Congressional option. That's a totally different thing.
As far as delegates to a Convention of the States, the provision of Article II Section 6 that forbids a federal officeholder from being a presidential elector would also apply to the position of delegate to such a convention. The supporters of the Convention of the States movement pretty much agree on that.
I think you are reading stuff into the constitution that isn’t there.
The States are free to send whoever they want to a convention of the States. If the President is a citizen of New York then he could obviously be chosen by the State of New York to be one of the delegates to a convention of the States.