We're first introduced to the Confession Dial during The Magician’s Apprentice as a Time Lord equivalent of a Last Will and Testament.
Both a Will and a Dial concern the disposition of things that a decedent leaves behind. For us resource-poor mortals, a Will usually focuses on the disposition of properties -- who inherits what, and the how and why behind it. However, that's not the only thing a Will can accomplish. A Will handles unfinished business.
Confessions are a way to determine unfinished business. Sins that have been committed but are still unatoned, mistakes that are still waiting for correction, testimony that can free the unjustly prosecuted, secret knowledge that must be passed on but must only be given to a worthy secret-keeper -- these are the sorts of things I would expect in a Time Lord equivalent of a Will and Testament. These are the sorts of things that a confession-gathering system can uncover.
The Confession Dial makes sense to me as an instrument that discovers and then advocates finishing a Time Lord's unfinished business. The Time Lord gets to discover for himself what part of his unfinished business he wishes to leave unfinished and unconfessed, and what part he wishes his executors and heirs to finish for him. He is given both time enough to remember everything and motive enough to confess every worthwhile confession. In the dial's normal usage, once he has confessed all that he can confess, the Time Lord embraces his final death and begins his afterlife in the Matrix.
The Dial can advocate finishing this unfinished business, but it can't act completely on its own to do that. It needs an executor, someone who can listen and then take appropriate action. It is inevitable that some of the Confessions themselves will be embarrassing, private and delicate things. It only makes sense the executor be a very close friend or family member -- someone the Time Lord implicitly trusts to have sound judgement around private and delicate things.
Missy's morals may be worse than suspect, but her judgement remains sound. She's a reasonable executor.
Unfortunately for the still-living Doctor, the Time Lord High Council itself is also a reasonable executor for a Time Lord's Confession Dial. Once the Dial was safely back in his possession, the Doctor had no problem with making that final Confession. He does not mind that the Dial knows. I suspect that, if he had Confessed inside the modified Dial, he would have been released but the Dial would then be in the possession of the Gallifreyan government. A dead Doctor probably wouldn't care but a still-living Doctor still has secrets to keep and still has reasons to mistrust that government.
If we can think of the Dial as a way for a Time Lord's final confessions to reach the right hands, then we can see how similar that is to a Will as a way for a mortal man's final wishes to reach the right hands. It's something a Time Lord can use in the hopes that his survivors will be able to fulfil his obligations and fix his mistakes, after he is no longer able to act on his own behalf.