You are asking why they would say true things.
No, I am asking why they would knowingly desire the approval of those who prefer "irredeemable" candidates. They would either have to lie a lot to get it, or pull themselves down to be more reprehensible. So, what's their strategy? Lieing a lot after telling the "one truth", or becoming more reprehensible themselves? Probably both.
Seeking votes is not like seeking approval in a social context. Someone trying to win a contested election desires votes for the purpose of winning.
You did: "The latter is probably true, but why would they say that" The implication of your comment is that politicians shouldn't tell the truth because that offends voters.