Is there a word for the phenomenon where explicitly stating assumed things, unprompted, makes you question their validity?

As you point out, the examples are of two different phenomena (awkward redundancy and stating the obvious), either of which might prompt you to question the truth of a statement or its implications depending on the social situation.

Rather than looking for a linguistic term that encompasses both (and possibly other) phenomena, such as hesitations and unusual word choice, I would look at the pragmatic value such utterances have in a conversation. They could be classified as one or more of terms below, the choice depending on how certain you are in your judgement, which might take additional factors into account, such as intonation and body language.

All of these nouns refer to signs, spoken or otherwise, that may affect the reliability of information, possibly revealing contradictions, untruths, information below the surface, and the like.

tell (n.)

A revealing gesture, expression, etc., that is likened to a poker player's tell

But his eyes darted fractionally to one side as he said it …; the classic liar's tell. — Stephen King

Broadly : SIGN, INDICATION

I talked to staffers who said that their bosses had two or three flights booked getting out of town, a big tell that there was not going to be a deal. — Kelly O'Donnell m-w

tip-off (n.)

WARNING, TIP

A telltale sign m-w

giveaway (n.)

An unintentional revelation or betrayal m-w

dead giveaway (n.)

Something that shows (the truth about something) clearly

The camera is a dead giveaway that you're a tourist. m-w