Hi moon. To confirm, The Drifter doesn't even offer to you? Have you unlocked the "Primed and Ready" triumph? If not, you have to get this triumph to start the Allegiance quest, which requires you playing a match of Gambit Prime and Reckoning. After you finish Gambit Prime, go speak to the Drifter who will give you a Weak Synthesizer which allows you to go into The Reckoning.
No offering. I had thought I did prime and stuff on this character , but let me double check.