I see your point about different cultures creating conflict because everyone has different backgrounds but I also think human personality causes conflict. One thing I like about Stamets is the guy is just a moody kind of guy who isn't always good with people. I would like to find out why he is like that but I do think in the end personality clashes is were most good conflict comes from. You do have conflict of high minded idea's which comes to play a lot in Trek but sometimes I just like the idea that someone is cocky and someone is shy and those personality traits create conflict. The writers do have to be honest with who their characters are. One good example might be someone not finding Tilly's jokey stuff amusing now that she is on a command track. Stamets might actually might not be a good chief engineer if that is the job he gets because he will have personal under him and he is basically being a bad boss even if he is smart enough to do the job. I'm sure their are many more examples you can work with.
I dunno. While I think that the TNG-era dictate that every plot of the week contains sci-fi elements was ridiculous, at the same time, you have to do something with the underlying setting - even if it's just using the setting as allegory, or introducing a sci-fi MacGuffin to examine some universal of the human condition.
For example, take The City on the Edge of Forever. The emotional core of the story is universal - it's the conflict Kirk feels between his love for a single individual and a sense of duty and the needs of the many outweighing his own. The story itself however is based upon a sci-fi MacGuffin however - time travel - which could not take place in a normal drama.
Or take DS9's Duet. It's a simple allegorical tale, which just as easily could take place in a historical drama shortly after World War 2. However, in reality it's intimately woven into the worldbuilding that both TNG and the first season of DS9 up until that point had done regarding the Cardassians and Bajorans. You couldn't just drop it into any old show and have it work. Outside of Trek, much of NuBSG falls in this realm as well.
In contrast, petty bickering could be on any show. By all means, include fights if they're about something topical to the series, or reflect fundamental differences of opinion. And I certainly don't want to go back to the TNG era where everyone always agrees by the end of an episode. But unless you get two actors with great "frenemy" chemistry it's something I'd really prefer not be played up.