Yep. In the context of what should be a very simple crystal structure, "complexity" is another word for "defect".

For decades, the marketing message was that less defects = better = more expensive. Apparently, when lab-grown diamonds came along, that had to be inverted: now, lab-grown = less defects = "less character"

The whole thing is such an obvious marketing exercise with very little to back it up (as evidenced by the extremely low resale value of diamonds)

Exactly, at the same price yes the natural stone will have more 'complexity' and be lower-graded than the lab one as a result.