Wine-dark seas & rosy dawns & many-storied stars

It really wouldn't have taken all that much shuffling and reassignment plotwise to add him in as a supporting character.

We would feel intrigued at the beginning of "Attack of Clones" when we learn that Count Dooku has left the Jedi Order after Qui-Gon's death. We could see Anakin and Obi-Wan briefly exchange lines about how they miss Master Dooku as well as Qui-Gon (there is already an exchange in the films where they state they miss Qui-Gon), and how they haven't seen or heard from him in some time now. Anakin could suggest that Dooku is hunting down the Sith Master; Obi-Wan could counter with how Master Dooku has simply returned to his life on Serenno, which he couldn't have as a Jedi Master, which Anakin casually calls unfair and he suggests that Dooku can do far greater good as a powerful count (a parallel to Anakin's marriage to Padmé and own Fall). Dooku being established earlier in the trilogy would better highlight how he and Obi-Wan went completely separate directions after Qui-Gon's death.

And again, the reveal that Dooku has Fallen would hurt so much more, if we had actually seen him be affectionate and righteous and wise. If we had any point of comparison for how Dooku's embittered desire for peace and justice has been warped into the pursuit of control and tyranny. It would hurt to see that formerly good man sentence Padmé to death as "just politics, my dear".

"This will start a war!" Padmé tells the man who helped her help her people once.

"I know," Dooku replies, with ominous satisfaction.

It would hurt to see Obi-Wan beg Dooku to stop this (a prelude to him begging Anakin in the next movie: "Anakin, please, I cannot lose you too!"), only for Dooku to attack and nearly kill him when Obi-Wan refuses to join him. It would hurt to see this grandfatherly figure cut off Anakin's hand, someone he knew and was kind to as a child. Seeing where Dooku fell from would also make everything about his fight with Yoda hurt more as well. We wouldn't have seen Dooku's struggles directly, offscreen in the time skip between TPM and AOTC, but this Fall would help prepare us for witnessing Anakin's Fall onscreen in "Revenge of the Sith", illustrate for us how power and grief corrupts, how the desire to take complete control and "start over" corrupts.

And all of this would also make Dooku's death in ROTS hurt more: to see Anakin execute an unarmed, injured man who had once been kind to him, who had once had good intentions a long, long time ago. We could have even had Dooku perhaps try to warn Anakin about Sidious, as the fear cuts through him as he realizes Sidious has betrayed him, only for Anakin to kill Dooku out of anger (Dooku is responsible for so much death, Palpatine reminds Anakin) just before the ruined man can finish speaking. Dooku's former goodness underlines Anakin's arrogance in thinking that his own fate will be any different.

The novelizations of the prequel films and other extended universe materials build up an image of Dooku's life as a Jedi and his Fall for us. We can assume and imagine a lot. We can retroactively apply knowledge gleaned from "The Clone Wars" with Dooku as a major villain. But ultimately, Dooku as a more sympathetic and emotionally relevant character is just not in the films.

When "Attack of the Clones" reveals to us: "Oh, no! Dooku has betrayed the Jedi Order and the Republic!" I think that most of the audience is like: "Gonna be real with you, chief, I have no idea who that is."

He's only been mentioned before once maybe? In Palpatine's office? Master Mundi assures Palpatine that Dooku is a good man (or something like that), but we have seen no evidence of this ourselves. This line mostly just becomes really funny on a rewatch, rather than poignant, because the prequel films audience only ever gets to see Count Dooku as a Sith Lord and rather underdeveloped villain. We don't ever get to see him be a "good guy" first. We're told but not shown.

The audience has no solid reason to care that Dooku specifically has betrayed the Order, as opposed to any random Jedi, because we haven't seen him before at all, much less interacting with any of our protagonists or establishing himself as an opinionated player within the story. Which is a shame! Because he has strong opinions that stand in interesting ideological conflict with so many other characters, generating fun and dramatic exchanges! He has direct connections to and parallels with other characters! He's potentially a really useful storytelling tool within these films, and his character just doesn't get used to that full tragic potential.