Sounds like you're conflating things. A popular theory among fans during the third season was that Kovich was the President, though I don't recall anyone thinking it was a secret. Regardless, the theory was put to rest once the fourth season arrived and actually introduced the President.
Watched The Examples. Another episode that stands alone whilst running along with the overall DMA thing. I like that the DMA is a manufactured thing. I'm already spoiled a bit, so don't worry, but I like being drip-fed the mystery. Tarka is a great character. Again, don't worry, I know he's a bad'un, but he lights up the screen. I think Hugh and Paul's relationship is as well handled as ever and I love how positive the depiction of a gay relationship is in this show. The scenes between them just chatting are beautiful and put me in mind of how I talk to my wife. That's good writing. Zora! That's very cool. I know we get a Zora episode later, so looking forward to that. Kovich is a bad ass! No crying. No pausing a crisis to hold hands. Maybe my favourite of the season so far and I feel invested in the story very much now.
I remember when I first saw the episode I didn't particularly care for that part. Mainly because at the time, all the talk online from the writers explaining how the DMA was basically the pandemic made it seem like they were drawing on the "COVID was created in a lab" conspiracy theories. Fortunately, by the end of the season the storyline goes in directions that makes it clear that wasn't what they were doing, but at the time, it was something that seemed rather glaring.
I tell you what was glaring. Rhys was given some lines and a job to do. A job which apparently he did very well, though we will never know because it occurred entirely off screen. His prominence felt forced and I was sure he was going to die. Shadows of Airiam. I'm also feeling underlying echoes of TOS beneath it all. I think S4 is DSC's most TOS like season so far. I can't explain why I think that but I do.
I didn't think he was going to die. More along the lines of, "Oh! They're throwing him a bone!" I don't get why people want to see more of Rhys. I'll be completely blunt: I don't give a shit about him. I really don't. Owosekun is interesting, Detmer is interesting, and Airiam was interesting. Rhys is just the dude in the back. I think he's there because someone decided, "We can't have the bridge be ALL women, plus Saru!" I say don't worry about it. He's a nothing character and we don't need token men just to have men. But anyway... I'm not going to spoil anything but you're on the right track. You'll definitely see it once you get to the later episodes of the season. I've said on here that this season has felt more Star Trek than anything made after TMP. I stand by that. It becomes downright Roddenberrian. I might've already said too much, so I'll stop there.
Nothing against the Wok-Choon, who is... fine? But I feel the same. I couldn't remember the characters name until Michael said it and he's been around since Season 1. I think he was on the Shenzhou? Quite an achievement to have remained so completely under the radar for so long. In universe, Rhys came across so desperate, like: Rhys: Captain please assign me to that job, because even though I've never done that job I'm sure I'd be good at it. Later: Michael: How did it go? Rhys: (offscreen) Great, all finished, no hitches at all! And that was that. It was very odd. Give Rhys something to do and then don't show him doing it. As you say, most likely throwing an actor, an actor who literally has not been given a chance to make an impression over 3 and a bit series, a bone. It was a very shit bone though. A brief interaction on the bridge and a short bit of ADR work in the back half of the episode. At least Patrick Wok-Choon gets to collect an easy paycheque. I'm already quite spoiled, so don't worry. It's little things that keep niggling at me with TOS feels. The whole bit with the Examples themselves struck me as a scene which would be easy to transplant into a TOS script, with Michael as Kirk and Book as Bones. It's a very strong season so far.
Although, the Enterprise's bridge in SNW is all women besides Pike and Spock. No, Rhys was not on the Shenzhou. The only one of Disco's current bridge crew to have been on the Shenzhou is Detmer.
There's a lot to unpack in Stormy Weather. I think it's my favourite episode of S4 so far. Zora is a fascinating character and I can't help feeling Roddenberry would have loved the whole concept. The stuff with Book and his Dad was beautifully done. Strong. One of the best DSC episodes overall I think.