There's a point where arrogance becomes straight gullibility. It's one thing if Dooku is so arrogant he thinks he can sidestep Palpatine's (very, very obvious) trap for him and be the one to lead the New Order himself (or whatever), but it's another for him to somehow forget that the Sith's
entire thing is betrayal at the first opportunity.
I mean, he knows
for a fact that Sidious is actually Chancellor Palpatine and that the whole Clone Wars was some astroturfed bullshit the two of them cooked up to allow the Sith to seize control of the galaxy...and he never thought to make like a video saying "Hey if you're watching this I'm dead and
Palpatine's Behind It All" or something?
Shit, why not just blurt that out when Anakin's debating whether or not to chop his head off? Your hands are gone, your boss is going full Dew It, you've got nothing to lose bruh.
Yeah that's the overarching problem with the PT - all the characterization is done after the fact in order to explain around Lucas' first draft writing, with the films being weirdly coy about Sidious identity. TPM straight up ends with
PALPATINE'S BEHIND IT ALL, but then AOTC is suddenly "Oooh wouldn't you like to know the Secret Sith Lord???? What could be their plan??????????", and then ROTS is like two hours of the heroes running around doing unrelated shit before having their minds blown that Palpatine is, in fact, behind it all.