I think, perhaps, the reason TOSs's social commentary seems unoffensive to us is because the message it was trying to get across is, for lack of a better term, "no duh" for a modern viewing audience.
For instance, a big issue was not just having a female bridge officer (originally Roddenberry wanted a female second in-command, but the network turned that down flat), but also an African American female. Kirk and Uhura were also TV's first interracial kiss.
Most of us today would look at that and think, " So what?" To us it's commonplace, or at least not something that's completely bizarre. To viewers in the 1960s, however, it was a major issue for those for and against it.
Having a Russian man on the bridge was also a big thing because TOS was shown during the height of the Cold War.
Many of TOS's storylines and alien interactions were analogous (to varying degrees) with the issues between the US and the USSR. The Klingon Empire was an alien stand-in for the Soviet Union. Unsurprisingly, the Federation was supposed to the United States, or at least NATO. Look at the starship designations: every name begins with "USS" which is what you see on American naval vessels. They retconned this and made "USS" stand for "United Star Ship" ("United Space Ship" was also used earlier on) instead of the modern-day "United States Ship."
All of this to say that TOS was probably seen as quite controversial and may have been considered "preachy" back during its initial run.
I think the big difference today is that during the 1960s, TOS had to be subtle in its approach. This wasn't necessarily because the writers or Roddenberry wanted that, it was due to stricter guidelines and censorship. Nowadays a show can bash viewers over the head with social commentary and get away with it, so the art of being intentionally or unintentionally subtle is no longer relevant.
For the record, I dislike the socio-political commentary you see in almost every show today, especially because it feels forced, like writers are using a checklist in order to seem progressive. I have nothing against social commentary or inclusivity, but it needs to fit the narrative.
Plus, I really wish some writers would stop changing the sex or race of established characters. This hasn't been an issue with Star Trek, but it has plagued many other shows, movies, and books. Sometimes it can work (Nick Fury comes to mind, but that has more to do with Samuel L. Jackson's inherent badassery) and other times it's just really, really weird (Flash Thompson or Mary Jane in the new Spider-Man movies).
A good way around this is to create a new character. The best example of this is Miles Morales, a young, black Spider-Man from a different reality. He's a new character, so that's cool, and nothing says Spider-Man can't be black. However, Peter Parker isn't black so he shouldn't be changed, and Marvel did an excellent job with this by leaving Parker the way he is and adding a new character.
Anyway... I've written enough and it's time for dinner.