It feels like they had half a system in place, honestly, with the explorers running research in the universities and museums.
The key issue is that the artifacts are a limited, global resource, and that the process of discovering them is not linked to culture. The way I'd change it is - you need a museum to run the research. Once the museum is in place, you unlock a project to look for artifacts. The project costs culture, instead of production, and the cost scales, same as with settlers. If a city is using its culture that way, it's not contributing it to your totals used for civics. Once the research is done, it spawns one artifact somewhere on the map, that's only visible to you (don't mind if it's permanently, or for x turns). You still benefit from having explorers spread over the map - you grab them quicker - but the artifacts themselves are effectively infinite. You can then tune the culture costs so that discovering 15 is about as time-consuming as researching the full tech tree.
ETA: an alternative would be making it so a museum only finds an artifact on its own continent, and costs are separate per continent. That way, exploration age colonies are rewarded, and conquering colonies with museums is viable counterplay to slow things down.
