What would Star Trek shows look like if the senior staff didn't go on all the away missions?
It's a classic, if somewhat exaggerated trope in Star Trek: The ships first officer, second officer, tactical officer, chief engineer, chief medical officer, and a random ensign beam down to an unsecured planet while some dangerous problem is either ongoing or likely to occur. The Doylist reasons for this are as obvious as the Watsonian reasons it seems so silly: these are the main characters who are supposed to get the bulk of the screen time, so they are constantly thrown into situations which real world commanding officers and department heads are generally kept well clear of.
But what if this wasn't the precedent established in TOS and continued in every subsequent series (including, to a slightly lesser but very real extent, Lower Decks)? What would a Star Trek show look like which still had senior officers who we are meant to care about and who still get significant development and screen time, but who aren't thrown into unrealistically dangerous situations on a regular basis? Could such a show survive telling stories without visibly putting those regulars lives on the line so frequently? Would it be viable to keep the focus on things that happen either aboard ship or in nominally safe situations? Alternately, could a show successfully develop a cast of lower ranking "away team" characters who get the "dangerous" screen time while keeping significant focus on the major decision makers on the bridge? And how could the shows manage such a visible separation between "expendable" and "not expendable" crew while maintaining that humanist, optimistic, everybody-has-an-equal-right-to-life ethos?
It wouldn't be an easy thing to pull off, certainly. But how could it have been done?
Reviewing your questions to consider, it's very hard for me to conceptualize a show that fits the description in your title. Most of Star Trek is heading down to the planet of the week. Given the choice between focusing on the away team and focusing on the crew operating the starship, I think I'd sooner follow the away team and consign the ship crew to being nameless extras.
What's interesting, though, is that I can think of at least two occasions where the people making Star Trek had similar doubts to the ones you've articulated here. First, when Roddenberry decided that sending the captain down was too dangerous, which led to the development of Riker as a character. Second, when the Enterprise writing staff decided that what Star Trek really needed were marines and came up with the MACOs.
So, while I can't really envision a Star Trek where the main cast is confined to the ship, I can envision a Star Trek where a starship's senior staff is distinct from a starship's MACO command staff and the main cast is split between the two.
In other words, we're talking about a version of TNG where Riker, Yar, and Worf are not Starfleet officers, but MACO officers. In this version of TNG, away missions are composite affairs: Geordi is still heading down if there's an engineering problem to solve, Crusher is still heading down to respond to a medical emergency, and Data is still heading down in case they need to win $12.5m playing craps. But Lt. Col. Riker is still in command of the the away mission and Capt. Worf is bringing up the rear.
The thing is, this changes the texture of your average away team-centric episode so little that all we've really done is... add marines to Star Trek. This will inevitably pull Star Trek in a militaristic direction and I don't think we've gained anything in exchange.
Closing thought. While writing this response I encountered something that surprised me: Major Hayes is only in five episodes of Enterprise. I suppose it's a credit to Culp's performance that I would have guessed he was in at least ten episodes had you put me on the spot and asked, but on the other hand, it's pretty telling that even though Enterprise kept the MACOs through season 4, they just became redshirts and the Enterprise writers never even bothered to tell us who their new commander was.
It's hard to imagine Lt. Col. Riker faring any better given the same constraints. You can give the Enterprise a MarDet, but if you're going to give them something to do on a regular basis that isn't equally or better suited for the senior staff, then you're writing a far more action oriented show than we're accustomed do.
I think the MACO example is a really great starting point though: have a few "away teams" that can beam down or beam over and take care of things. The exception being when circumstance requires a specific officer. In Enterprise during the Xindi conflict, MACOs handled certain off-ship actions. I think a similar situation in another franchise is Stargate SG-1/Atlantis, where the bridge crews never leave the ship and instead utilize teams (such as SG-1 or AR-1).
Starfleet claims they are not a military organization, but beams down with phasers (or phase rifles). Why not get a special team who is experienced militarily and instill first-contact procedures, diplomacy, and, if the need requires it, a science personnel who is trained to defend themselves.
With as many red/yellow shirts who have died, it makes sense to not endanger your senior officers who would have died without plot armor. For their credit, the MACOs were a step up from their plain security personnel.