Any decent renderer, including Mapnik used on osm.org is able to do this. Of course, automatic label positioning can give rather strange results compared to a handcrafted map.
This is a very interesting discussion. OSM is focused so much on precisely mapping features, while regions are inherently imprecise. An example of an extraordinarily fraught regional moniker is the Midwest of the US. There are continual (mostly good natured) debates about what constitutes this region and most of it focuses on precision at the level of states (e.g. Which States Are in the Midwest? | FiveThirtyEight). It seems like the best way to map these types of regions would be to use some sort of raster type heatmap (not that I’m proposing this for OSM).
Acutally I do not see any requirements towards OSM in mapping regions. There is the appropriate tag and there are several options to use it. Draw a precise area or a roughly estimated one an give it its name. If you are not even sure about the area, place a node somewhere in the estimated center and name it.
If you search in OSM carto for a region which is mapped you will well find the result in form of a marker and if you like you can also visualize the area (if mapped). Yes, the region name is not rendered in carto and I think this is ok due to the impreciseness of region areas and other reasons mentioned earlier in this thread.
So whoever wants to map regions, go ahead … 
The question risen by @ jumbanho how to determine the area of a certain region is another issue but from my point of view this is not an OSM problem.
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