I've completed A Blade Reborn on 2 characters on my account. I've claimed the MoT book on both characters, but neither would trigger the completion. This is the only MoT I need to finish the book and it has been the only one to not complete since MoT came out for year 2.
To be sure, I double checked the abandoned quests kiosk and did not find the Blade Reborn quest there (or any of its precursor quests).
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I see you have 9 deleted characters on your playstation account. Is it possible that one of those deleted characters are the one that obtained the exotic sword(s)? If so... that's the issue. You will have to obtain the quest again on one of your current characters.