No, not necessarily. PAOs do essentially "come from the ground" if one wants to be picky, but if one also wishes to be picky, fatty acids and alcohols come from nature, too, don't they?
I would suggest that the historical definition for "synthetic" when it refers to base stocks is based upon the notion that said base stocks can be formulated in the lab by a variety of means without petroleum products. However, that doesn't preclude petroleum base stocks or byproducts as raw materials either, now does it?
The chemistry of alkenes is well studied and there are plenty of reactions and uses. There are enough topics in that subject alone for fodder for multiple PhD and Masters theses in the field of chemistry.
I have no problem with marketing Group III/III+ as synthetic, though I am well aware of the definition of technical synthetics, and those are Group IV and V for a reason. I prefer to let IUPAC member organizations and IUPAC itself regulate definitions in the field of chemistry, not marketing departments, advertising review boards, or legislators.
That being said, I have no problem with a Group III being called a synthetic. I know the difference between a "synthetic" and a technical synthetic just the same as I know the difference between "work" and the product of force and displacement. But, the unemployment office isn't going to be issuing a directive to me on what units to use in a a calculation of force times distance.