I think the agreement itself was drafted to specifically avoid having to be considered a new treaty in a US context. So from a legal standpoint, I think it's incorrect to say "this is a treaty and must be ratified by the Senate". I will heartily agree with you that perhaps that shouldn't be the case, and also agree that we need a functional system of government where these machinations aren't necessary to pass policy, but I'm not sure there's a constitutional requirement here per se.
Other countries such as Brazil had the treaty ratified by their senate.
Arguing that we now have a treaty that allows us to make other treaties without following the Constitutional process ... well.