Being among the first adult SF series has always been a Star Trek standard. However in later years the modern series seem to have bent over backwards to remain inoffensive and family friendly.
This did not happen to Twilight Zone, Galactica or Stargate.
When and why did this happen?
I think you're misunderstanding the sense in which "adult SF series" was intended to apply to TOS. It didn't mean a show that was for adults
only -- just one that wasn't aimed primarily at children the way
Lost in Space and the like were. It meant a drama on the maturity level of contemporaries like
Gunsmoke. And yes, Roddenberry did push the envelope in the depiction of skin and sexuality.
But despite that, TOS was always meant to be appropriate for children to watch, at least older children. In its first couple of seasons, it aired in an 8:30 time slot, early enough that a lot of kids would still be awake. It was always popular with children, and when it went into syndicated reruns -- often in afternoon time slots -- young audiences were a major component of its fanbase. (And don't forget, ST:TMP was released with a G rating.) There are many of us, myself included, who are such loyal fans because we discovered
Star Trek very early in our lives. And indeed, many people in the '60s and '70s saw ST as a children's show because that's what they assumed all science fiction was. ST was meant to be a departure from that trend, but a lot of people didn't get the memo.
Still, I agree that the later shows did tend to be more staid and conservative, less willing to take risks, than TOS was. But that was less about being a "family show" and more about being a highly profitable institution rather than an underdog. The more money something makes, the more reluctant its beneficiaries are to take chances that might hurt the bottom line.
Still, there are exceptions. TNG's first season was the only one that Roddenberry was personally in charge of, thus it tended to have a lot of sexual content -- Data and Tasha in "The Naked Now," the hedonist planet in "Justice," and the like. TNG got somewhat more staid later on, but DS9 was willing to be more risky and risque. The same-sex kiss in "Rejoined" was highly controversial when it aired and some stations refused to show it. And
Enterprise often played up the sexuality, but in a clumsy and juvenile way.