I think the issue that I, and the others, have with using the reboot trilogy as a starting point is maybe tonal. To a degree, it had "grittier" more in-your face horror trappings, and the supernatural was a bit downplayed--like I'm tripping over how to adequately describeit, because its not absent by any stretch, or even hidden.
Like maybe 2/3 of the way through 2013 or Rise or Shadow, you encounter immortal/relict/preternatural warrior peoples. You're scarcely a quarter through the original game and are fighting a T. Rex, and things go up from there. In Legend, you very quickly see a magic stone and have a flashback of your college class being eaten by a shadow monster. Its so much more. Its more in your face, and bigger and bolder. Its... I guess survival horror versus ridiculous pulp.
Plus, the survivor trilogy has always been Lara is stuck, here. Rise and Shadow have unconnected prologues, but then you go from one interconnected area to the next. Older Tomb Raider games were globetrotting adventures all around the world. The Andes to Greece to Egypt to Atlantis. A hidden tomb underneath the Great Wall of China, and a quest from Venice to a sunken ocean liner to Tibet to get the key to it. Peru to Africa to Tokyo to the tomb of King Arthur. As opposed to the old gulag right next to the ancient Byzantine city, or an impoverished South American town to the secret hidden Mayan village next door, to the Spanish mission a block away (exaggerated, yes, but thats kinda what it felt like).
Like, the anime could in fact, be a grand high adventure connected to the Survivor timeline by ignoring the tone it went for. But the chances of it being so are higher if they just did their own universe.