Okay so what if Palpatine like genuinely cared about the Republic? He was trying to preserve it, but it was collapsing. The Jedi don't care, "if it must fall then it's the will of The Force". He falls to the dark side in his desperation, looking for any way to change fate. He uses his new Sith powers to manipulate people and events to his designs.
Anakin's mom dies in the first film, and instead what causes the start of his fall to the dark side at the climax of the second film is Dooku, or rather Palpatine, using some ancient Sith artifact to simply show him the future. Padme will die and he will have to die too to fulfill the prophecy, it's in The Force. He flips out. He doesn't want to be the chosen one anymore and realizes The Force has only brought suffering to his life. Obi Wan tries to convince him that it's just a trick and gives some meaningless platitudes, but the seed of doubt is planted.
Thus in the third film Anakin's recruitment makes much more sense. They are both trying to do the same thing. Fight the tyranny of The Force. Palpatine to unnaturally prolongue the reign of the Republic, Anakin to unnaturally prolongue Padme's life and alter his own destiny.
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