That's the way it was sold. But the fact is that while being 18 means you can hold a gun and shoot it where you are told, it does not follow that you have the capability to judge what should be shot at.
Btw, the draft ended about that same time, but the voting age was lowered anyway. Which tells me that this selling point was just that, and not the real reason.
I would like to raise the age but you are wrong on both counts. GIs between the ages of 17 to 21 have a lot of responsibility and for some of them the judgement required of them will be higher than what they encounter in their civilian jobs during the rest of their life.
The draft ended almost two years after the voting age change.