The ending makes perfect sense -- it's just been a matter of us all trying to explain it, in a clear and thorough way.
I'd put it like this:
Liz could manipulate time-space because part of her (the tip of her pinky finger) was in one universe, while the rest of her was in another. It's like she's standing on the border between worlds, one foot on each side, able to see both sides.
To try and regulate this power, the siphon was created inside Monument Island, as a way to restrain her. Liz describes it as a "leash." Once the siphon is destroyed at the end of the game, she's "off the leash," and her power reaches its full potential. She can now see all of the infinite sets of timelines in the universe... including all of the ones in which Booker becomes Comstock.
To prevent Comstock from ever happening, Liz has to create a PARADOX, because the universe "does not like its peas mixed with its porridge," as Lutece put it -- or in other words, nature will correct any paradoxes by obliterating paradoxical timelines from existence.
So, Liz creates a paradox: She drowns Booker before his baptism. This creates a paradox because if Booker is dead, Booker can never become Comstock, and if Booker can never become Comstock, Comstock can never steal Liz, and if Liz is never stolen, Liz never receives her ability to traverse time-space and kill Booker in the first place.
The universe sees this and goes, "PARADOX!" And then obliterates each and every timeline where Booker becomes Comstock.
All that remains, are the timelines where Booker rejects baptism. What was once a "variable" -- an element that can change, in this case to accept or reject baptism -- is now a "constant," like the coin that always comes up heads when the Lutece twins meet Booker again at the Raffle Fair and ask him to flip the coin.
That's an important concept to understand. There are constants -- elements that always work out the same across all timelines -- and variables, or things that are different depending on the timeline. Elizabeth, by creating a paradox, forced the universe to take the "variable" of accepting/rejecting baptism, and turn it into the "constant" of rejection.
And so Booker, while still in debt, will be able to see his daughter grow up. And hopefully things will work out for the best.