The thing about this is that it's less tied to a time period and more to choices made by the creators. TOS, TNG, DS9, VOY, ENT, and various Streaming Trek shows have all had varying amounts of "frontier-eque" vibes. I think people go back to TOS as having the most because, well, it probably did, mostly thanks to the fact that the overall universe/canon/whatever weren't as established (hence a lot of inconsistencies across those 3 seasons) so it feels different. By S7 of TNG a lot more of the universe was established and DS9 really turned the Federation/Alpha Quadrant in to a much more "real" place, so we approach the franchise that way now. SNW might take place chronologically before Captain JTK but the creators and audiences are so much more aware/knowledgable about the canonical universe.
This is actually one of the reasons I like some of the retcons/changes SNW makes. Scary XenoGorn are new and cool and add a new, very intense dimension, for example.
Yeah, but i think it's more about atmosphere and tone than the lore. All the shows since S2 never got it right imho.