I think it was always implied she'd be captain at some point. I don't think it was actual lie. I never once assumed she wouldn't be.
The musical chairs with captains is why I think she'll officially get it by the end of the season. They can't keep doing that. I know some people want Saru as Captain but look at Saru's actual screen time. It's not the highest in the cast even though he's a main character. All that make up will take its toll on Doug Jones. He's likely not going to want to up the hours he's in it and they're likely not going to be willing to do that either, no matter how much some people want an alien captain. There are many logistical reasons why it's not reasonable on a regular basis.
They're not going to skip the lead actor for the position and want to deal with the blow back that would come from doing that. First lead black female but she's going to be stuck with the commander rank the entire series!? There is already an issue with them starting off the two black leads in Trek as commanders instead of captains like all the other leads. Don't think that isn't noticed by people either. Yeah it's fire they're playing with here if they do this wrong.
I'll be laughing if Kirk got command of the Discovery for S3. It might actually be fun if they can get the right charismatic actor to pull it off. By that point, he'd be about 5 years away from his Enterprise command.
They should probably hold off 'Captain Burnham' till the end.