The laws and treaties must be made in accordance to the constitution, so it is supreme in that the other two parts are subject to it and not the other way around.
While laws and treaties must operate within certain parameters outlined by the Constitution states are not constrained by the Constituion if they want to extend more protections than the Constitution provides.
If the founders were of a mind that the Constitution was the "Supreme Law of the Land" it would have been quite easy to confine Artilce 6, Clause 2 to exactly that. They didn't and they didn't for a reason.
So when you say the Constitution is the Supreme Law of the Land (period), you contradict the Constitution.
But that doesn't surprise me, there are so many people who simply ignore the plain wording of the Constitution it has lost its meaning.