I think the distinction is that the user synonym feature fixes the issue of Wanikani never being able to account for all possible correct ways of phrasing a meaning (and things like users who would like to add synonyms in their first language if it’s not English), while still locking the user into having to accept that a missed answer is a miss no matter what the reason was (user scripts aside).
For better or for worse, the lack of this feature in the main site seems to be something the designers of the website feel strongly about, to the point that it’s listed as something they are unlikely to consider in the “About the feedback category” post:
If it helps at all, personally I think of the discipline you mention as a reason to keep the takeback feature limited to user scripts, as I found it helpful to instill that discipline when I was a new learner still figuring out how best to learn a language. Misses, even ones I disagreed with, reminded me that the only consequence was seeing that item a little more often in the SRS piles, and that seeing an item again wasn’t something to be upset about. Which is a perspective that I think in some small way helped prepare me for confronting reams of words and kanji in books and the like.
That’s just my take though, and plenty of people get lots of use out of the takeback script as well. 
So hopefully its existence, at least, provides the best of both worlds.