It is actually not easy for a synonym to get approved without moderator intervention. In order for a user to vote to approve a synonym, in addition to meeting the reputation threshold the user must have a score of at least 5 in that tag. We only have 139 users who meet the reputation threshold in the first place, many of whom are no longer active on the site. And four approval votes are necessary for a synonym to be approved. Many of the lesser used tags probably don't even have four users who meet both requirements for voting. And even the few users who can approve suggestions probably only visit the synonyms page rarely.
This problem has been discussed at length on the network-wide Meta site, most notably in this post, and several suggestions have been made. However, it does not seem like the system will be drastically improved in the near future. If you want to draw attention to a particular synonym proposal, you can mention it in Chat or in a dedicated Meta post.
(As an aside, you don't need to go to the Help Center to propose a synonym. You can actually propose it from the tag page itself.)