Anakin’s massacre of the Tusken raiders is also the wrong kind of step into evil, from a character perspective. We’re supposed to see Anakin becoming Darth Vader, and Vader is in the main a cold-blooded villain

I've said for 20 years that Anakin's fall, in the movies, comes off mostly as an ass-pull. The scene where he kneels to Palpatine and Palpatine literally seems to dream up his iconic villain name on the spot has always felt like exactly that. "Oh well, Windu fell out a window, guess I'll be evil now!"

That's probably why these fash weirdos are pumping the ROTS novelization rather than the movie itself, as I understand it sheds light on Anakin's internal narrative in much more detail. I've always had the sense that Lucas was trying to say something about the (sadly still relevant) trend of American school shootings with both the Tusken massacre and the infamous youngling slaughter, but we don't get enough time with Anakin in the movies to gain much idea as to why he would reach directly for mass violence. If you're just going by the movies, the idea that he's still carrying unresolved trauma from being enslaved carries nearly all that weight. And if that's all you've got to go on, it reflects pretty badly on the Jedi not helping him with that for the 10-15 years he's with them. The movies do seem to admit that the Jedi Order is a failing institution, but this conflicts with the shiny happy optimism of Luke growing into a Jedi in the original trilogy. Rian Johnson tried to square this circle later, but obviously he didn't convince everybody.

If anything, Anakin's interactions with Dooku are the most convincing bit. Hotheaded apprentice gets in over his head -> later matures just enough that he has to be pushed into taking murderous revenge. That points toward the violent-cop narrative you suggest, but we don't spend nearly enough time with that storyline for it to breathe. My favorite fan works posit Vader as essentially being Palpatine's slave, and thus psychologically beaten down to the point of near collapse. fialleril on AO3 is one of the most inspirational in writing this sort of stuff.