Because it was only 3 members doing combat, everything else contributed by say Germany was do to secondary treaties. For example staging, hospital use, and maintenance in Germany would be covered by basing treaties. Yes quite a bit of this is wrapped up with NATO as a whole, but quite a bit also isnt lots of ifs, ands, ors, and buts in the language of it all.
Turns out alliances and treaties can be complex and esoteric things at the best of times, we figured that out after WW1. And yes only three NATO members had participated in the 2003 invasion of Iraq and one of them (Poland) pulled out. If you bring up Australia they arent part of NATO. Iraq was an unjustified clusterfuck ya dont need to lie about it to make it sound worse.
Also intelligence agencies dont count, half the time intelligence is in a region its to make sure it doesnt need them.