Is necessary to take the Cabal legacy flaw to be initiated in the first station at character creation, even if you were the one to kill the Former Primus?
Generally Cabal Legacy is for Mystery Cults rather than the House mysteries.
Is the character initiated at gauntlet, or is the game starting some years post gauntlet?
If you've got some years post gauntlet, I'd say no - you can just have picked it up as part of normal pre-game progression.
As a SG, I probably wouldn't accept a newly gauntletted magus who had already killed the Criamon Primus and Initiated the Path of Strife. Firstly, it's usually a non Criamon who kills the Primus, and they're usually chosen beforehand (unlike the Primus' successor), so it feels unlikely the knife would have been wielded by a Criamon apprentice (or even a non-Criamon apprentice who later joined the House). Secondly, the House tends to discourage people from going down the Path of Strife, so having been initiated into it that early seems unlikely.
Killing the Criamon Primus during their ritual is a very particular backstory and I probably would not allow it for fairly young magus but also remember they are not of House Criamon and they are unlikely to have a score in Enigmatic Wisdom and even if they do they must pay for that as a separate virtue and are unlikely to have a very high score in it.
[Edit to actually answer the question asked] But, no, Salutor is correct above with respect to House Mysteries generally, Cabal Legacy is not needed for House mysteries with characters starting after Gauntlet.