Swapping a synonym doesn't actually do all that much, and certainly not as much as you think it's doing. Literally the only thing it does is change the synonym record so that the master and source tags are swapped. It doesn't retag any posts, and doesn't change any past synonyms.
That tag doesn't "show up" in that scenario because a) it is not a synonym that points somewhere else, so we don't show a stub regardless and b) it does not have any questions tagged with it. There is no case where we show a tag that is the master of another tag but doesn't have any questions, because that's not really a case that's even supposed to exist.
There's another step you want to take after swapping a synonym, and that's a tag merge. On the synonyms page where you performed the swap, a merge link would've appeared immediately after, because it detects a bunch of questions with a tag that's supposed to be remapped to a master tag. Do it. Completing the merge will solve all the issues you think went wrong with the swap by:
- Creating the tag, if it doesn't exist.
- Replacing the "desexing" tag with "neutering" on all the questions.
- Moving any synonyms that point to "desexing" to now point at "neutering" instead.