But he then brought that Thawne back to the present with him and let him get away. So that Thawne should have never been stuck in 2000, become Harrison Wells, and did all the other stuff we know. So it's inconsistent.
Exactly. The time remnant should have been a place holder to become Harrison Wells, while the primary Thawne is released from his destiny, from a point in his personal timeline before the events that set Eddie into removing him are begun, no longer bound by past casual events. This Thawne should only have memories up until the night he tried to kill Nora, and whatever he learned from Barry during Flashpoint imprisonment. The only thing that makes sense is that through the speedforce, he was able to aquire his primary memories / consciousness.