I think both you and pyuser583 are correct to a certain extent. The stated reasons for overturning Roe were because of the tenuous basis as a privacy grounded position. On the other hand I completely believe that Roe was such a politically charged issue that the judges voted according to their allegiance, even though they are not supposed to have such allegiances. Everybody would have been surprised if any of the judges had decided differently to how they did. So, while there may have been a legal argument to be made, I don't think that particular issue was decided on those grounds.
The reasoning behind Roe was generally regarded as tenuous even by the justices that supported it. Overturning it was required to defend the government’s Constitutional authority for agencies like the FDA, which was undermined by inconsistencies introduced by Roe v Wade. Eventually those judicial inconsistencies come home to roost.
tl;dr: Roe being overturned had little to do with privacy and more to do with protecting specific regulatory powers from being unconstitutional using the same reasoning introduced in Roe v Wade.
Removing such decisions from Federal purview was an elegant solution to the problem, with the practical effect of deferring all such decisions to voters at the State level.