Why did Palpatine spend so much time in the prequels politicking his way into being in control of the Senate

This is an issue you have with rise of skywalker not ROTS

Because his hold on power is much stronger if it appears to be perfectly legal.

In regards to ROTS, it's called "Playing the long game." Many villains end up failing at their evil plots or being a flash in the pan, because they move too fast with their plans, and have no backup plan in case their main one fails. Palpatine had actually been fulfilling the plan his master, Darth Plaegus had started a generation before. It's one of the few times I've seen a villain actually succeed in their plan, and he ruled the galaxy for at least 20-some years before Luke convinced Darth Vader to betray him and throw him into the reactor.

In regards to TROS, (which I do not consider to be Star Wars canon), bad writing, studio meddling, and abrupt changes were made to the script because Iger and KK didn't like or know who the villain was that JJ and Lucas came up with. Halfway through filming, they demanded JJ change the villain to someone everybody who had watched previous films actually were familiar with, rather than taking a chance with a guy nobody knew about, save for those who had watched the Clone Wars cartoons.

It was a very poor and half-assed decision to include a Palpatine clone in this new movie.