Earlier quoted context omitted.
>infamous ECDSA nonce Why "infamous"?
It's more properly called 'k'. It's really a secret key, but it has to be unique per-signature. If an attacker can ever guess a single bit of the nonce with probability non-negligibly >50%, they can find the private key of whoever signed the message(s). It makes ECDSA very brittle, and quite prone to side-channel attacks (since those can get attackers exactly such information.
This doesn’t seem right. Why wouldn’t someone guess a bit 0, see if the recovered message makes sense, and if it doesn’t, then try bit 1?
It would make the entire scheme useless no? Am I missing something?