Synonyms are a great way to go, but the full, proper name is what should be the parent tag.
Tags are standardized. We have conventions, and there's no reason Star Trek should be an exception now that we've got the character limit to allow it to fit the full titles. A great example is a-song-of-ice-and-fire which has the child synonym of the oft-used "asoiaf".
There's also network-wide conventions with tags, such as them being in English (on the majority of stacks), which is the standard, supported language of the network. Similarly, we should move towards tag names being the standard names of the works, where possible.
One of the issues with these fan-based abbreviations is that, by necessity, the current parent tag has to be
star-trek-ds9, even though it's already a synonym of
deep-space-nine. "Deep Space Nine" is enough to find the work if all you know is the title of the show. "DS9" needs help from a synonym, the prefix "Star Trek" or someone to know the abbreviations.
Lastly, synonyms are supposed to be for the "common" terminology, while parent tags are meant to be for the full, proper term. That's even the guidance of the synonym page:
Incorrectly tagged questions are hard to find and answer. If you know of common, alternate spellings or phrasings for this tag, add them here so we can automatically correct them in the future. For example, suggest “bike” as a synonym for bicycle, or “sock” for socks.
In this case, "tng" is "bike" and "the-next-generation" is bicycle, and tags should be "bicycle".
Fears of people not picking tags isn't relevant, or even more than a wild guess it will happen. It's a bit insulting to think that someone trying to find "deep space nine" won't use it because the tag looks too long, even though it clearly contains the title of the work they want. It's also not realistic that our many reviewers and users won't add in missing tags when they see them. (And we're far, far from having too much edit or review work in queue. Many users often complain that we don't have enough, as they're trying to earn badges or reputation.)