Because Star Trek: Discovery is not actually about the senior staff of the ship. It's about a particular set of officers aboard the ship, some of whom are part of the senior staff and some of whom aren't.
When the particular identity of the USS Discovery's executive officer becomes relevant to the story, it will be established.
I would suggest that the focus is on the main title characters, and that thinking it's either primarily about the senior staff or about the lower decks, as though it must adhere to some binary, is a mistake.
There's no reason to say the show "should" show the XO or the chief engineer. The show has no obligation to focus on senior staff.
I think the main characters have all earned major promotions many times over by now. I'm not bothered by the show not depicting, for example, the chief engineer, but I also don't think it would be at all lacking of drama, vindication, or catharsis if Paul were to receive that promotion.
I don't think the show "should" show any position due to rank or title and I agree that Discovery wants to be all about a specific subset of the crew regardless of their rank or position. What I object to is what eschaton pointed out above: the show wants to focus on character X, so it contrives a reason (or just jumps to that character for no reason) to get them involved. Or, for no reason ignores the logical character who should be involved in the situation. Why have Burnham leave the bridge twice without identifying her XO? There was a whole storyline last season dedicated to trying to find one. Why leave the role heavily implied but not stated? That is the perfect, in story, reason to highlight it. That is the perfect time to give whatever character who was chosen last year their moment in the spotlight - to bask in a moment where we see who was deemed worthy to be second in command. Why call to Stamets for an engineering solution? He isn't an engineer. He knows some stuff about engines and power systems because of his work with the Spore Drive, but he isn't probably even in the top 10% of crew on the ship in terms of general engineering knowledge. The sole reason he is involved in that scene is because he is a main character for the show. You wouldn't call up Culber for an engineering solution just because he is a main character and hasn't gotten a scene recently.
If the show wants to have X character in a scene they need to 1) write problems for which that character already has the skill set or knowledge to solve or 2) show that character getting the skill or role for which you are writing these problems, either through training or promotion. The whole reason that Trek shows tend to involve the standard "classes" of characters (engineers, scientists, doctors, commanders, diplomats, etc.) is that the shows tend to write engineering, science, medical, leadership, or communications issues. It's the show being stuck using existing Trek tropes but not wanting to use the appropriate characters that causes the mismatch. If your chosen main characters (from a story perspective) don't have the skills or roles that allow them to logically solve the problems your writers are writing, you need to write different problems.
This is a self-contradictory statement. The Federation's ongoing recovery from the Burn is the very essence of what it means to be in the 32nd Century.
But the show isn't interested in the science fiction part of the equation. They only care about the fall of the Federation storyline (which is agnostic of time period; they could have set it after the end of the Dominion War, basing it on a longer-term collapse of infrastructure from the war, or a contagion that spread due to poor food/medical accessibility), they don't actually care about the 32nd century setting. They don't care about "what future technologies might address/eliminate older Trek challenges?" They don't care about what future technologies might exacerbate or mitigate the impacts of the fall of the Federation or it's reconstruction. They don't care about the science fiction, only the drama portions of the setting. The same thing happened in Enterprise; they had all the same gadgets (phase pistols and polarized hull plating), and problems "solved" the same way - except in rare instances - and you could have placed a large portion of Enterprise episodes in any other timeframe with no appreciable differences. That is what it means to not care about the science fiction part of the setting.
Why? A setting in which technology never fails is a setting devoid of dramatic tension. They made it very clear: The transporters did not work because the impacts from the extremely large amount of debris that was impacting on them at high velocity due to the gravitational anomaly had damaged the Heisenberg compensators.
I agree, if there were no problems that couldn't be easily addressed, or if the technology always succeeded, there would be no tension. There is no problem with clarity (other than the problem as stated makes no logical sense). The problem is the execution. Practically the only new technological development that has been shown with any appreciable effect on how things operate in the 32nd century is the ubiquity of transporter technology. Characters don't even need to walk anywhere on a starship anymore if they don't want. But the first major incident of the season and the transporters fail in exactly the same way they would have an any other Trek setting in the past, and in a way that is illogical and contradictory to precisely the way this technology has been shown to operate now. It is almost like they chose the one new technology and said "what if we revert it to the way it was in the past, so we don't have to worry about it being different anymore? So, we can just use all the old tropes and storylines and won't have to think of anything new or actually consider the future setting at all. It will be so much easier this way."
Don't agree here at all. Hell, most of the complaints against DIS are that it's too big of a departure from ST tropes.
I think the main criticisms have been in 1) the visuals of the show: elements looking different than they have in the past for no appreciable story reason, just that new designers are involved and because they can, and 2) differences in tone or a focus on "darkness" and "grittiness" which goes against the core elements of Trek - belief in a better, optimistic future with a focus on improving humanity itself, and 3) general lack of quality (individual mileage may vary).
I would love for (and do love when) Discovery does something new (no more "dang, all the transporters are out again"), but when they do something old I would like them to do it right.
Just thought, a cool story beat would have been something like:
1: the unprecedented gravity anomaly prevents the transporters from working, and prevents comms/drones from operating
2: the 32nd century transporter technology literally hasn't had a single failure in 200 years, so no 32nd century Starfleet officer has ever had to do an EVA, Starfleet literally doesn't even have EVA suits, and all the 32nd century officers are ill-equipped to address the situation.
3: because the Discovery is from the past and Burnham has done a lot of EVA missions, that makes her uniquely qualified to go on the dangerous rescue mission.
This would have been more logical and character/setting appropriate. But, Discovery isn't really invested in doing those things, so we get a non-sensical transporter failure and the captain deciding she should go on a dangerous EVA mission despite being the captain.