Why would the Jedi not go public with their knowledge of the Sith, especially Count Dooku?

Question

This question was inspired by one about whether the public in the Galactic Republic knew Count Dooku was a Sith Lord.

At the start of the movie, the Jedi still thought very highly of Count Dooku even though he left the Jedi Order.

MACE WINDU: You know, M’Lady, Count Dooku was once a Jedi. He wouldn’t assassinate anyone, it is not in his character.

KI-ADI-MUNDI: He is a political idealist, not a murderer.

By the end of the movie, they knew Dooku was a killer.

The Jedi learned that a Sith Lord named Sidious controlled the Senate from Count Dooku.

Count Dooku: What if I told you that the Republic was now under the control of a Dark Lord of the Sith?

Obi-Wan: No, that's not possible. The Jedi would be aware of it.

Count Dooku: The dark side has clouded their vision. Hundreds of senators are now under the influence of a Sith Lord called Darth Sidious.

The Jedi didn't believe it at first, but by the end of The Attack of the Clones, the Jedi decide to watch the Senate.

Kenobi: Do you believe what Dooku said about Sidious controlling the Senate? It doesn't feel right.

Yoda: Joined the Dark Side Dooku has. Lies, deceit, creating mistrust are his ways now.

Mace: Nevertheless, I feel we should keep a closer eye on the Senate.

Yoda: I agree.

Once the Jedi were forced to rethink their views on Dooku, why didn't they tell the public?

They could have told the public, "Hey, that Confederacy of Independent Systems you want to join? It's controlled by a man who has ordered the deaths of thousands of people. He's not just a man who left the Jedi Order, but has become very corrupt who will use lies and deceit to destroy the Republic."

They could probably back up their words with video footage of the Battle of Geonosis; video footage showing Count Dooku watching the slaughter of Jedi knights within the arena. (Assuming the clones has body cams, and that their attack ships had cameras.)

To my best knowledge, the Jedi never told the public that Count Dooku had gone bad, and that a Sith Lord controlled the Senate. By going public, they could have:

• undermined confidence in Dooku's leadership,
• dissuaded planets from joining his CIS,
• and gotten the public to turn a skeptical eye toward the Senate.

All of these outcomes would serve the Republic and undermine the Sith Lord's plans. So why not go public?

Edit to add: To my best knowledge, the Jedi didn't learn until episode, “The Lost One” (The Clone Wars, S6:E10), that Count Dooku is not just a fallen Jedi but a Sith Lord as well. So once they learned that, they could have also told the public, "Count Dooku is a Sith Lord. History says that each time a Sith Lord tried to create an empire, he ended up causing the deaths of millions or billions of people."

Answer 1

Well ...

1. General public doesn't know much about Sith. More educated ones do know, but it is ancient history. It would sound like crackpot conspiracy theory without proof ...

2. ... which leads as to another thing. Jedi do not have concrete proof about Sith involvement. Only hearsay from various un-reputable sources like Count Dooku himself. They didn't even have lower part of Darth Maul, nor his lightsaber.

3. They only learned that man called Tyranus who created Clone Army is actually Count Dooku late in the game (S6E10,as OP mentioned) . Note: not Darth Tyranus, just Tyranus. They suspected he was Sith Lord, but again no concrete proof.

4. Jedi Order from prequels is well ... arrogant, prideful, and basically weak ( both in Force and in brains :) ) . Yes, Count Dooku actually told them everything they needed to know back on Geonosis. They never had intellectual courage to think about it seriously. Indeed, during whole prequels they were trying to sweep things under rug, win Clone War, find and kill Sith, and go back to their usual routine. Very few of them considered any changes to corrupt Republic system . Ironically, those who did turned to the Dark Side (Dooku and Anakin)

5. Citizens of Republic (majority) do not want the truth. Excluding few volunteers, most of them are generally disinterested in fate of of the Republic. They have Jedi Order with their "magic" powers, they have Clones, they have "representatives of the people" in which they do not trust much. Telling them that Dooku is a Sith, and that there exists even more powerful Darth Sidious would not change much . Like Jedi, they just want war to end and to continue with their usual little lives .