And what about the disutility of saying "yes, I did burn her alive" and this potentially leaking back to his supporters ?

You mean if he actually said it as clearly as that to Lucius Malfoy?

Almost nobody believed Lucius, not even his own supporters in the Wizengamot, as has been mentioned in the story already. Only the Death Eaters seem to have believed it, perhaps because Voldemort believed it.

It's different if The Boy Who Lived testifies to the same effect (that Dumbledore told him he did it) infront of the Wizengamot.