No, what you say is correct, but you don't even need to bring entanglement into it at all: moving faster than light is the same thing as moving into the past (in some reference frame). This is why information can't propagate faster than light.

The kind of time travel that I'm talking about here is not merely sending information into the past but sending yourself into the past, that is, sending your body into the past. But that's not possible because your body is on the most fundamental level made of entanglements, and entanglements define the arrow of time.