Looking at Heretic's source code, at the very least, it appears that there is code preventing the Ring of Invincibility from respawning, but it's not in quite as obvious a place.
However, looking at Hexen's source code it appears to be a whole different story - I actually don't even know what to make of this. https://github.com/videogamepreservatio ... #L878-L901
It suggests that there are actually respawn "states" where it respawns said artifacts after a period of time as defined by their wait state.
How deep does this rabbit hole go? Well, I went and looked up the actual state the artifact goes in (in-world) when it is picked up. https://github.com/videogamepreservatio ... #L411-L431
What conclusion to draw from this? It comes back after 10 minutes, plus a short amount of animation time. Furthermore, the other artifacts as defined near the links I gave above, have different periods of respawn time, and if not specifically defined, do not respawn at all.
I kind of figured fixing this would not be as simple as adding flags. Some more evaluation and discussion is merited for how to handle this.