You can try this:

position = GeoPosition[{54.612947, -5.900967}];
region = GeoDisk[position, Quantity[2, "Kilometers"]];
mp = GeoGraphics[region, GeoBackground -> "VectorLabels"]
Cases[%, _Text, All]

I can explain that difference:

When the first argument of GeoGraphics or DynamicGeoGraphics contains several point-locations then we show a map of a region that contains all those locations, with a bit (say around 10%) of padding. The same thing happens when the first argument contains something with a finite size (say a disk, or a polygon).

But when there is only one point-location we cannot infer a notion of size, and we default to a radius of 10 miles. This is what happened when you restricted the region to 0.5 km: GeoEntities only found one entity, and therefore GeoGraphics and DynamicGeoGraphics did not know how to compute a map radius. Adding GeoRange -> region or GeoRange -> Quantity[0.5, "Kilometers"] would indicate the scale you want, and solve the issue.

And yes, we don't yet have entities for streets.

As Jose mentioned, WL does not have entities for streets. You could use a free reverse geocoding service e.g.

Import["https://geocode.maps.co/reverse?lat=54.612947&lon=-5.900967", "RAWJSON"] // 
Dataset

It's possible to get more street names in the maps by zooming enough. You can do this by clicking the + sign on the lower right of the DynamicGeoGraphics maps, or by adding GeoRange -> region or GeoRange -> Quantity[0.5, "Kilometers"] to the DynamicGeoGraphics call.

You can also use GeoGraphics instead of DynamicGeoGraphics, and then use one of the vector backgrounds, for example with GeoBackground -> "VectorClassic", that may give you access to other labeled elements. Try for example:

GeoGraphics[region, GeoBackground -> "VectorClassic"]