To some measure your thoughts are addressed by the defence of "good character".
If a person (I'll assume man, though the law does not), be accused of "kiddy fiddling", one defence he may maintain is that he of a person of unblemished good character. He may bring character witnesses, he may bring evidence to show that he has himself children, that there has never been any suggestion of impropiety, that he is a pillar of the community etc. To say to the jury "How unlikely then is it that a man such as I, held in high esteem, no shadow previously having crossed my reputation, trusted by all, how improbable it must be that I would do such a thing as I am accused of".
But (I think - correction welcome) that if a defendant does maintain a defence of good character, then the prosecution may bring in evidence of previous convictions. (" The defendant claims to be of unimpeachable moral character - how then does he account for his three previous convictions of kiddy fiddling").
But the prosecution may not refer to previous transgression unless the defendant claims first that there is none.
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