The issue of overlapping or redundant tags is something I’ve also been thinking about, and I consider it one of the persistent problems within OSM.
(Please take a look at this post regarding this matter. Of course, I imagine there are many similar cases across OSM as well.)
I suspect the post above reflects similar concerns, but in general, overlapping attributes on a single map feature are always a headache. In fact, it’s quite common to see multiple different tags being applied to a single object.
That said, I’d like to clarify two specific points in response to your suggestion:
First, as I’ve emphasized several times already, in the case of Vietnam, I believe the ceremonial gate tag should only be used for features that clearly qualify as Tam quan.
As @osmuser63783 also pointed out earlier in this thread, simple gantries or portals—even if they share a similar shape—should not be tagged as ceremonial gate.
(Of course, I do understand that in countries where religious symbols are viewed with suspicion or negative sentiment, structures that originally served ceremonial purposes may now function purely as wayfinding signs.)
Beyond Vietnam, this kind of structure—clearly influenced by Chinese architectural styles—also appears commonly at the entrances to villages in various Southeast Asian countries. (Example)
Second, if the primary purpose of the structure is not propaganda or advertising, but rather to serve as a physical object such as a portal, gantry, or gate, then I believe it should be tagged accordingly, based on its original function.
The presence of political or promotional messaging on part of the structure is, in my view, a separate issue.
In such cases, your idea of using an additional, separate tag to reflect those secondary elements might be a good solution.