FormerlyHemlock
Hero
Well, it says "at the end of each of its turns," so I guess that would mean at the end of the round, the target gets another saving throw to break out. Wait, that doesn't make much sense. I don't know if these concurrent turns can be systemically reconciled with all the content that assumes cyclic turns.
That oddity is orthogonal to the choice of initiative system. You can get the same effect under PHB initiative, if the Hold Person spell goes off on the target's own turn. E.g. he attacks, then moves towards another target so he can threaten them with opportunity attacks, which triggers someone's readied action to Hold Person anyone who approaches; bad guy fails his save but then gets another save at the end of his turn, so he didn't actually lose anything by his first failed save.
No matter what the initiative system, I'd just treat that as "at the end of each of its turns except the one where the spell is cast." I.e. you never get multiple saves on the same turn.