There are different types of treaties, but as an example, no one in the Danish government is directly responsible for trade treaties. That's all handled by the EU. Member countries of the EU aren't permitted to even negotiate trade matters with non-EU countries. Similarly much of Denmark's defense policy is guided by, or constrained by if you prefer, its membership in NATO.
I'm not saying the Prime Ministership of Denmark is unimportant, but I do believe people, particularly people in positions of power, tend to overstate their importance. If I recall, Belgium, which has almost twice the