The concepts aren't really that "high." Stripped down to the basics, a lot of the plots could be found in contemporaneous dramas of the day Which is what Roddenberry was going for. He wasn't about High concept, but relatable situations with a veneer of Science Fiction.
I think you might be overstating this; all fiction obviously hinges on an ability to relate to the characters, but high concept ideas absolutely are central to Star Trek. There's no real-world situation that echoes being held captive by an alien child who thinks he's a naval officer, or having to decide what to do with a 1960s pilot who's seen your futuristic spaceship, or stopping a war fought between computers, or fighting a genetically engineered superman in an engine room.
Sure the plots revolve around basic human states like being in danger, being in love, standing up for friends, making moral stands, and showing empathy to the other, but those are universal ideas found in all fiction. I'm not sure I'd describe "City on the Edge of Forever" as a relatable situation; it's rooted in common human experiences like love and sacrifice, but the situation is completely fantastical to the point where it reads as essentially mythic, which I think is part of why it's barely aged.
Thing is TOS isn't a show about Kirk in a cave set. It is a show about feelings and identity with an emphasis on dialog. Kirk (or Picard for that matter) monologuing is a trope for a reason. Spock coming to terms with who he is several times a season is also what TOS is about.
This might be where the differences between TOS and DSC start to emerge - Kirk monologues tend to be obvious things like "war is bad", "hate is bad", "trust is good". Picard monologues generally go the same way even if they're dressed up more.
Does Discovery speak to universal experiences in the same way? I'm not sure it does, in part because it's always a bit hesitant to really say anything specific - the plots themselves
could lend themselves to that (Culber being killed then reformed by the mycilleal network, for example, feels very classic sci-fi) but the presentation, the choice of language, the specific psychological concepts the characters are written to model, and where the focus is placed all feel much more contemporary and culture-locked to me than TOS typically does.
The first season's plot, from what I remember, basically involves a series of bizarre events that are thinly-drawn - Lorca is from the MU and wants to "Make the Empire Glorious Again" and then dies by falling into a death-field, T'Kuvma is leading a Klingon movement that seems to be ethnonationalist but is again vaguely defined, Tyler is Loq and has a memory of being assaulted by L'Rell as he was surgically altered, Mudd appears for a while, Starfleet is torturing a tardigrade but then lets it go, Burnham is caught in the middle of all this and generally reacts with shock and dismay. There's a war going on which ends when L'Rell and Burnham acquire a superweapon. I'm just not sure any of these threads resolve in a way that echoes TOS' universally-applicable model of "here's the plot, here's the resolution, here's Kirk saying 'war is bad'"; instead it's a very MFA-y idea of "the feeling is foremost, the plot is secondary, and we must reject easy resolutions", which I think again ages like milk, is a very transient model of how television "should" be, and reads as muddled.
Barring the post-season-one sexism (which really is the big thing that cripples TOS), the only thing that dates TOS in the same way is when Kirk occasionally slips into a very mid-century notion of equating "hard work" with "freedom", like in "The Apple", and tellingly those are parts where modern viewers tend to laugh/cringe.