From my research, it appears that if a user has a gold badge for a tag A, and a question is tagged with a tag B which is a synonym of A, the user won't be able to bindingly close as a duplicate questions that use the tag B. Furthermore, if the same user makes a non-tag edit or unrelated tag edit to the post to get the system to swap the synonym tag B out for the canonical tag A, the system still won't allow them to close-hammer them, since as far as it knows, the user was the first one to add that tag in and the mechanism to prevent gaming the system kicks in (even though, strictly speaking, they aren't gaming the system).
The most common case where this ends up becoming an issue is when a synonym is created without merging the tags. When a tag synonym from B to A is created, this means that all current and future attempts to use tag B will automatically be swapped out for tag A, and when any question that uses tag B is edited, it swaps it out for tag A. However, existing questions that use tag B and aren't edited after the synonym is created will still continue to use tag B: merging the tags, i.e. remapping previous uses of tag B to A, is a separate process. This is to provide a means to reverse the synonym mapping in case a mistake is made, as merging will also rewrite posts' histories to use tag A and is as such impossible to reverse.
The support part of the question: is my research on this correct? The only source I could find documenting this behavior is this post, which reports another case where this ends up becoming an issue: when a question is migrated to an international (non-English) site meta, the system adds in the discussion tag in English even though the required tag is different on that site (it's in the site language), and users with a gold badge for the actual required tag aren't able to bindingly close these migrated questions.
The discussion part of the question: assuming my analysis of this behavior is correct, should this be fixed so that gold-badge users with a badge for tag A can close questions tagged with B which is a synonym of A? One argument for this is that since the tags are the same concept, and synonyms can be created by normal users (who've posted answers in the canonical tag) while merging requires a moderator, this should be allowed. One argument against this is that this allows for hammers even in case of incorrect synonym pairs, and a moderator should finalize their merge before allowing users to hammer them, but I think incorrect synonym pairs are rare because normal users who make them must have posted answers in those tags. Which outweighs the other?
