It does not, the image processing is done by GeoServer, not by GeoWebCache, which merely takes the images and stores them on disk (well, it might take a larger image produced by WMS and slice it into tiles).
There are multiple topologies to consider:
multiple GeoServer instances with integrated GWC
one stand-alone GeoWebCache with multiple Geoserver on the ...
You need to use this URL https://gis.lmi.is/geoserver/gwc/service/wmts?Service=WMTS&Request=GetCapabilities&Version=1.0.0
For a WMTS, you need to replace after the base URL (all the url before ?) the string with the following Service=WMTS&Request=GetCapabilities&Version=1.0.0
So, in your case, https://gis.lmi.is/geoserver/gwc/service/wmts?...