I really don't think that's the case. From what I've read, grey hair appears thicker because usually grey hair comes with age, and less sebum production also comes with age. The lower sebum and oil production results in coarser, drier hair texture which results in greater hair volume. This is offset by natural age related thinning however. If sebum and oil was the cause, then people wouldn't suddenly start going bald in their 50s and 60s when their skin has become much drier.
My hair is already fine, even in my donor zone, so I dread to think how much thinner it can become!