For what it's worth: there's a 'real' version of Journal 3 and per the Love God's entry, the potions are explicitly temporary and if there isn't some ndrlying affection or attraction for them to work with relationships induced on them will crash and burn the second they wear off.
Anyway, I guess now I can explain the thing: Right before Bill Cipher is killed, he starts spouting off gibberish. Play the scene backward, however...
https://www.youtube.com/watch?v=anXDF2dr5kg
"Axolotl" being, per supplementary materials... Basically, the closest thing there is to a God in the series. Makes Bill and Tme Baby combined look like ants.
In
Dipper and Mabel and the Curse of the Time Pirates' Treasure, a Choose Your Own Adventure Book released in 2016, one of the possibilities is Dipper and Mable encountering the Axolotl, who takes the form of a giant, majestic axolotl, and it confirms that it
did reincarnate Bill, but in such a way that Bill would become a better person who would eventually atone for his crimes. Alex Hirsh has stated that while the book in general is non-canon, the twins meeting with the Axoltl
did occur.
...and if Gravity Falls and The Owl House took place in the years they both came out, then King would have hatched around the time Bill died. This, combined with the fact that Alex Hirsch used basically the same voice for the both of them...
However, this connection was explcitly denied... And requires ignoring the fact that King was self-aware in his egg and his egg would have been laid
millenia ago.
Also, a blink-and-you-'ll-miss-it shot shows that The Owl House takes place in 2022 so Bill would have been dead for 2 years when King hatched.