No, that is not why. It's because the US has two policies, domestic and international, and the application of treaties on domestic law is up to the federal government to execute. If the Congress chooses not to do so, there isn't a domestic law broken and so the states aren't bound.

So actually, if they had separate sovereignty, the states would have to sign the treaties themselves and would potentially be more bound to them, not less.

No, the United States formally withdrew from the Optional Protocol to the Convention concerning the Compulsory Settlement of Disputes 14 years ago. The cases alluded to in the parent comment (Sanchez-Llamas v. Oregon, Medellin v. Texas) came after the US withdrawal from the Optional Protocol.