Those Freemasons claiming some Templar legacy were just LARPers making shit up to look cooler than they were, the real successors to the Templars (as in, they were literally just OG Templars with a rebrand and a smaller white cross slapped on top of their original red one) were the Portuguese Order of Christ. These guys were actually pretty cool and I have never read about them being up to any subversive shenanigans in the vein of the Jesuits or Freemasons, instead they were mostly involved in the early Portuguese colonial ventures under Prince Henry the Navigator & Vasco da Gama (both explorers were members of the order, Prince Henry was even their Grandmaster) or in fighting Muslims in Morocco.