Aside from the episodes and general tone that I pointed out. There were other episodes that had controversial elements.
Some people are pointing out the sexual nature of early STNG. This is true, whatever you think of Gene Roddenberry, he did inject that into the episodes for a reason. Some people think it's enlightened, some just think it's perverted. I'm sure he wanted to put in more than he actually got away with. On facebook, I see a lot of people calling all of Star trek a "family" show when mentioning the lack of censorship upcoming in Discovery, and they are worried about it..but Gene didn't see it as just a family show.
Related to this, there were 3-4 episodes involving analogies to reproductive rights, whether it be clones or immaculate conceptions. Despite being a less than stellar outing, I always agreed with the viewpoint of Outrageous Okana..shedding some light on the old fashioned gender bias and possessive nature of men in women's reproduction. Again, these were less controversial because of the sci-fi element but they were there.
The Ferengi were possibly a failed foil for the crew in STNG, but I think Gene really wanted them to be an example of capitalism gone wild..as many saw and see America. The Last Outpost turns into a almost a treatise on how humanity overcame such thinking and the Ferengi hardly get a word in edgewise..but the Ferengi are so alien, has nothing to do with us right?
I always found The Chase interesting because everyone balks at the idea that enemies could be related to each other genetically when "clearly" we look so "different". This is exactly what happens if you tell people humanity evolved in Africa. I'd venture to say a huge percentage of the USA does not agree with this (I believe something like 40% don't even believe in evolution).
There are also quite a few episodes where direct combat is avoided, almost with Picard seemingly bending over backwards to avoid conflict. I think this was a central message in STNG, one which was slowly lost over the rest of the other Berman era sequels. Sure we all like action-adventure and TOS is very much that, and most of us like a good space battle or two, but this is really one of the things that makes STNG unique in all of ST and TV. It's an ideal that many of our primitive 20th century brains have trouble holding on to (apologies if you are a millennial). I think this frustrated many early viewers of STNG because Kirk would have probably shot first and asked questions later.
Maybe the "Outcast" was a little late in the TV landscape, but even now huge numbers of fans do not want homosexuality discussed at all in Star Trek. it's a family show and you can't expose kids to deviant thoughts don't you know. The way it was presented made it more digestible to viewers so perhaps it's not as controversial as it should be.
I felt some of the latter seasons of STNG were commenting on the anti-science of the world and especially as the political climate in the US took it's first turn to the dark side with Newt Gingrich in 93-94. The Gingrich Republicans were not as bad as the do-nothing idiots of Congress today, but they were the start. Eventually in 1995 Newt abolished the science advisory group OTA which has made it far easier to ignore facts and has fostered anti-technology and science in the US.
https://io9.gizmodo.com/a-key-reason-why-u-s-politicians-dont-understand-scien-1575132934
RAMA