Denmark joined because the US invoked Article 5 after 9/11. It was more about making sure they're protected if they ever need it in the future, so I'd say it was absolutely in their interest. This was the first time Article 5 was invoked, not going would've killed NATO and their own security. The invasion of Ukraine has solidified their reasoning.
As for article 5, that might explain the second gulf war and/or Afghanistan, but certainly not the first gulf war, Libya or Syria.
Whenever Uncle Sam says “jump”, you’ll see all the small NATO states lining up to ask “How high?”.
No it wouldn't.
The Article 5 invocation resulted in https://en.wikipedia.org/wiki/Operation_Eagle_Assist and https://en.wikipedia.org/wiki/Operation_Active_Endeavour, that's all.
>Whenever Uncle Sam says “jump”, you’ll see all the small NATO states lining up to ask “How high?”.
Consider that small NATO states want to partake in this stuff because it offers their forces an opportunity to gain experience.
Like how Denmark have joined pretty much every U.S. war of aggression in recent history, despite having no interests at all in Afghanistan or the middle east.