Totally agree with the first part. That was my mistake. They would indeed drop to 0 in round 1 and their next turn might be in that round, depending on where they and the foe were in the initiative order.The character's next turn is the turn they next take after dropping to 0 hp. So, if they're dropped to 0 in round 1 and then their turn comes up later in round 1, they make their disadvantage death save at the end of that turn.
All the spell description is spelling out is that it happens at the end of their next turn, so they still get to act normally on that turn, and then make the save.
Not totally sure at the second part of your post. The wording of the spell doesn't say they make their next death save "AT" the end of their next turn. It says they don't make them until the end of their next turn.
Once the character reaches the end of their turn, their turn is over, so they can't act, even to make a death save. That's my read on it anyway, your milage may differ.