You can only get 1 patreon core per season (per server). If you already collected the patreon core in EGS HQ, then you can’t also get one by the 7th fragment. It’s one or the other, but not both
The on in EGS HQ is a convinience, so you technically don’t have to get to the 7th fragment to get the core.
The one I did get before, I did paid for (patron master). It should be said visible on the website that nobody will get one inside the game if he did buy one with being a patron. That would save a lot of time.