Complete may mean something different from person to person but my definition is: competent in all ranges of fighting,striking,trapping,grappling,weapons, and also the spiritual side of martial arts. I'm not a complete martial artist yet but I'm striving to be a complete martial artist. What's your definition of complete? Also by complete I don't mean you know everything already and have nothing left to learn Its more about goal.
I’m sure it means different things to different people, as well as to different schools of thought.
Let me give you a "for instance”.
Say you teach. And you’re teaching a particular stye/system/dojo specific whatever. But you know a lot of Martial Artists and frequently work out with them. Say you learn a really good way to do something. And it’s better, much better, than what your particular group currently does. Do you teach it as an alternative? Or just let it go?
To me, if you don’t teach something you KNOW to be better, the only complete in you is you’re a complete chump.
I know if I thought my instructor knew a tactically better way for me to do something and he didn’t teach it to me for whatever reason, I’d sure never trust him again. And that ain’t good.
I think everyone should train in what you mentioned in your OP. "Striking, trapping (or how to take advantage of someone else trying to utilize trapping), grappling, weapons (and I mean modern weapons used in todays age) and the Spiritual side of Martial Arts.
To me, if you’re teaching things that involve defense, escape, survival and the ability to seriously hurt another human being, if you don’t teach at least some form of the Spiritual side of Martial Arts, then you ain’t got squat.
I taught in a city dojo of predominantly young men easily influenced by the world around them. To NOT include the Spiritual side of the Arts, would be completely irresponsible.
As for me being a complete Martial Artist, I'm lucky if I can find my gi pants on laundry day.