This is an argument about a continuum. There is no duality between hard and soft sci-fi except in people's head.
Trek inhabits a mid-point where Star Wars is your space fantasy and something like Interstellar or The Martian is textbook hard sci-fi.
Yes, exactly. Science fiction is not one style, but a whole spectrum of styles. Generally, a given author or creator will favor a specific approach to SF, including a specific degree of "hardness," but a collective work like a TV franchise, passing through numerous hands, will tend to encompass a variety of different approaches. Star Trek and Stargate are both TV franchises that have sometimes tried for scientific literacy and plausibility and sometimes indulged in more fanciful ideas, so they're hard to pin down and can best be regarded as "somewhere in the middle." Star Wars onscreen has never aspired to be anything other than a sword-and-sorcery fairy tale, though I gather some of the books have tried to ground it in more of a military-SF mode and apply some logic to it. Gene Roddenberry's Andromeda started out as a hard-SF space opera, but when the original developers were let go, their successors turned it into a nonsensical fantasy.
The problem for me, as I've said, is that mass-media SF tends to be way too heavily slanted toward the "soft," fanciful end of the spectrum, and thus doesn't have the same variety as prose SF. The norm is to be completely fanciful, and shows that even reach the middle of the spectrum are exceptions. Shows and films that lean toward the "hard" end are exceedingly rare, although there seems to be a recent and encouraging shift in that direction.
Yup. Compared to the mass of nonsense that constituted most screen SF back then -- compared to the likes of Lost in Space or Space: 1999 or the original Battlestar Galactica -- Star Trek was extraordinarly plausible. Its plausibility was like its feminism -- limited and backward compared to the best ideal, but significantly ahead of where most of its contemporaries were.But when Trek debuted, there was nothing like The Martian on TV. The closest were those Disney Man in Space documentaries. That is why it got the reputation it did back then, and it would not be fair to just retroactively bash it on this basis.