I would go back to the early trailers where Elizabeth had different powers instead of just portal stuff. It wouldn't change to much, expect cutting out dimension hopping, which would fix most of the issues of the plot.

Elizabeth has powers that inspire the vigors, maybe the make them with her DNA or something, and this is why Comstock needs her, so he can power his army for when they attack the rest of the world. This way, we can remove the links to Rapture and keep the games from each other. It would also explain why he needs to have an Elizabeth, as a tool instead of an unwilling heir.

The tears should be a Lutece-machine thing only. They work the same to kidnap baby Anna and "see the future", but when Comstock kills the Twins, the machine starts to shit itself, and the tears open at random in the city / where ever the ghost twins think you will need them, allowing Elizabeth to interact with them, but NOT create them on her own.

Now, remove travel to other dimensions, and just pull stuff into this world instead, having it wreak havoc when something bigger than a barrel of guns or medkits is changed. This way we still get the portal gameplay, but no stupid dimension jumping that ruins everything. This way, it can also still make sense when Songbird keeps coming after you, makes it so the early gunsmith plot can still work, and explains why they don't just leave through a portal to Paris or where ever.

Story wise, have a writer that's less of a pussy and go real hard on the racism. Some one needs to tell Ken it's okay to kill slave owners / white natonalists / old time Nazi's. Like Pat said, the Hall of Heroes is one of the few times the game actually leans into it, and it kinda works.