What I would’ve done:
The Jedi would’ve been a small religious order, their numbers somewhere between 100,000 & 1,000,000. They would’ve served as diplomats, advisors, and during the Clone Wars, as tacticians (they wouldn’t have fought on the battle field; that’s stupid). The general public would’ve had very little knowledge of the Jedi, and even less exposure to them.
I feel like them having a temple on the Galactic Capital planet and them fighting in public battles doesn’t fit with the what the Original Trilogy tells us (that people don’t know who they are and don’t believe in The Force). I’d prefer that when they interact with the public in the prequels, they use aliases and that the public doesn’t know that they’re Jedi and isn’t aware of their powers.
Anakin would’ve been inducted into the Sith religion by the Sith spirits on Korriban. Palpatine wouldn’t have been a Sith at all, but an independent dark side sorcerer.
Empire’s End implies this. Palpatine is the one who turned Anakin to the Dark Side, but he could still have gone through the trials and got knighted on Korriban like Exar Kun.