Star Trek has not given more weight to emotions than all other SF franchises, nor is it weird to do so.
- Star Wars: "Fear is the path to the dark side. Fear leads to anger. Anger leads to hate. Hate leads to suffering."
- Babylon 5: "what I wanted to do with making this show, was in large measure to examine the issues and emotions and events that precede a war, precipitate a war, the effects of the war itself, the end of the war and the aftermath of the war. The war is hardware; the people are at the center of the story." —J. Michael Straczynski
- Battlestar Galactica: the development of Cylons who can have emotions is crucial to the plot from the very beginning, and their emotions (as well as the humans') are important drivers of the story.
- Honorverse: the main character is empathically bonded to a "treecat"; in later stories she develops a general empathic talent
- A.I. Artificial Intelligence: it's the whole damn point
- The Foreigner Series has humans sharing a planet with aliens who do not feel love or friendship but have an equally powerful emotion called man'chi driving them that is similar to (but not the same as) human loyalty and lies at the core of their politics. Misunderstanding this caused a war when the species started to interact and they have strictly regulated their interactions since.
Need I continue? Basically, making emotions important happens naturally in most "soft" science fiction stories, and most SF has been at least partially "soft" since the 1970s - Star Trek can be seen as rather avant garde in that respect. Basically, the overwhelming view is now that SF is about people as much as about spaceships and rayguns, and emotions are what make people tick.
Another perspective: the three examples you cite where emotions are given "weight" in Star Trek are really gimmicks that are used in all kinds of SF and fantasy stories:
- Vulcans and Betazoids are aliens, and if you want an alien race to not just be humans that look strange, you need to make them somehow fundamentally different, and making their emotions work different is an easy and effective way to achieve that (and one that does not require expensive special effects). See my last example for a similar case.
- Data's "emotion chip" - that robots/androids have no emotions but are curious about them, or that complications ensue when they get them, must be one of the most worn-out tropes about robots out there (going all the way back to the Tin Man of Oz). I doubt there is any franchise where robots play a significant part that hasn't done this at one point or another.