That would be the case if one brother held all the titles and then passed them all on to the next brother. But what Henry did was award all his titles to all his sons (and, tellingly, refused to actually allow his eldest to actually have any power despite having the title) and then whenever one was removed from the succession (i.e. by dying) each son was forced to surrender all their titles and pass them on to the next son - at least with the prime roles, anyway. In a primogeniture case, each successive "oldest surviving" son would just start accumulating titles through succession, or acquire them at once, but wouldn't lose any. What Henry was doing wasn't primogeniture, it was "ranked-by-importance". As a standing member of the chain of succession, John would claim superiority to his nephew.
The problem is that when Henry died it went to the oldest son, Richard. You don't see John given Normandy and/or Anjou, you see Richard getting (in right of his father, since Eleanor's still alive and kicking) Normandy and Anjou and England - same as Richard's own son is getting unless Richard develops a soft spot for John for some reason.
So for the succession, I'd say it was definitely primogeniture. That Henry in life played games with empty titles doesn't really give John a legitimate position.
But it would have a great deal to do with whether or not anyone supports him as a legitimate claimant as opposed to just an ambitious and unpopular prat.Besides, all this is moot as, as I mentioned, the meddling of Philip Augustus would mean that, valid claim or no, John would eventually be manipulated into open rebellion during which he would claim the crown of England, anyway...