Daystrom Institute

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".