So I found the answer myself. I opened up the file geodatabase with ArcGIS and saw that it indeed contained raster datasets. So the answer is: Even the new QGIS completely ignores raster datasets when opening a file geodatabase. You wouldn't even know whether or not the GDB contains such data if you are not able to open it with ArcGIS. Very unfortunate.
This should get you started with reading in all of your feature classes, then you can perform a similar operation to what you've included in your question.
path<-"C:/Users/GIS Data/Some Folder/your.gdb"
st_layers will list all of the names within ...