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?
I’d argue that, in some ways, Deep Space Nine is the answer to your question. For the most part, DS9 did not utilize the “away team” concept much at all. Now, if you are asking more broadly about the effect of putting our characters “in danger”, than I suppose you could argue that all of DS9 was an “away mission”, but I think the dynamic was significantly different.
With respect to TNG, I suspect showing a wider diversity of crew on away missions would have heightened the feeling of the Enterprise-D as a “university town”, with a range of experts in different fields, but where the senior staff are seen – not as less expendable – but rather as generalists, or perhaps even more like “philosophers” (in an old-fashioned sense of the term), who must take in information from a much wider range of sources and figure out what to do with it.
Dramatically, however, I think this would have made TNG even more “talky” than it already was. Without the senior staff going planetside and seeing the strange new world for themselves, I think we would have that much less emotional involvement with the “extra of the week” doing the exploring instead. Could it have worked with, as you suggest, a subcast of “away team” characters? Perhaps, but I think you would have needed to remove some of the existing cast, or reimagine them significantly – I don’t think TNG could accommodate too many more regulars. (A rebooted TNG where Geordi, Worf, and Tasha are the “landing party” crew could be interesting, but would be very different from what we originally had.)
That all being said… I’ve long felt that Star Trek was at its best when it told stories that could be told as stageplays (or could be easily reimagined as stageplays). A TNG without away teams would work very well as a stageplay, and could serve as a way to focus the writing: the story has to be compelling through the dialogue and acting alone, and can’t lean on the tropes of the “dangerous away mission” or the “mystifying abandoned alien planet”.