The other dumbest thing about Obi-Wan's quest:
So, the entire galaxy spanning war (as far as the vast majority of the galaxy knows) is fought for four years between the Republic and the Separatists. The former led by Chancellor Palpatine, the latter by Count Dooku.
In truth, the whole war is a sham, and the two leaders are actually in cahoots. Dooku's rebellion is actually on the orders of Palpatine and they're secretly in constant contact throughout the war, coordinating their battles. This is such a tightly kept secret that even the top brass Inner circle on each side never gets a hint of it (Jedi generals and council on Palp's side, Separatist guild leaders and generals like Grievous, Ventress, etc on Dooku's side).
The story is so well planned out that Palp starts it in motion ten years earlier with a smaller rebellion on Naboo, to gain sympathy for the eventual Separatist cause. It's all so elaborate and meticulously planned that it fools the whole galaxy - billions upon billions of people!
So why in the possible hell does Dooku casually tell Obi-Wan the truth on the eve of the whole thing kicking off? He figures this loyal, council-seated Jedi whom he just met should be told Not only that he's a Sith, but also that there's another Sith leading the other side. 🤷🏻♂️
Over ten years of planning went into the most audacious, far reaching, history altering scheme ever, but at the last second, Dooku decides to just let the biggest lynchpin secret of the whole thing slip to some rando current Jedi he meets!
And THEN, even stupider, Obi-Wan goes back to Yoda and Mace and tells them, and they blow it off! They tell him ah forget it, Kenobi, Dooku is probably just making shit up. Then they spend the next four years fighting for Palp against Dooku, getting untold numbers of civilians killed, and it only dawns on them four years later at the end of the war that maybe they're both Siths on the same side!
It makes zero sense for Dooku to spill the beans, and even less for Yoda, Mace and Obi to just forget about it for the entire war! (Except that Lucas just really wanted the second movie to have a "Join me and we'll rebel against Palpatine" scene to mirror ESB.)