If a new tag synonym is proposed that I would be able to vote on (i.e. it would show up on my suggested tag synonym page), then please have it show up in the StackExchange™ MultiCollider SuperDropdown™ to alert me.
The Stack Exchange™ MultiCollider SuperDropdown™ is designed to show persistent information that ideally pertains to you. It doesn't show you what might be interesting to you, but what has been addressed to you recently.
Synonym postings aren't addressed to individual users. It's more of a side administration task, not unlike checking for questions in need of close votes or 48-hour closed questions that may need to be deleted. These aren't things that should be reported in your Inbox of things addressed specifically to you. These are all things that can be addressed by someone else without even needing you to step in, in fact - imagine getting notice of a bunch of new synonym proposals overnight only to step in and see that they've already been approved or rejected.
There's something to be said for having some manner of tracker to ease the job from the current method of just checking into the suggested tab every so often. But I don't agree that our global inbox is the place for it.