First, it's not about the legend (GetLegendGraphic), it's about the map (GetMap).
Here's what happens:
We ask the server what layers it has and what styles they use:
The server responds that it has many different layers, eg, it has a layer named 'Radiometrics2015_ternary_KThU' and this layer supports the single style named 'Radiometrics2015_ternary_KThU' (the name of the style coincides with the name of the layer). The relevant fragment of the capabilities document is in my post above.
According to the WMS spec (OGC 06-042, available here), the GetMap request has to include the STYLES parameter and this STYLES parameter has to convey the desired style for each requested layer. We are only requesting data for one layer so we are talking about a single style. The style for a specific layer can be either (a) one of the styles supported by the layer mentioned in the capabilities document, or (b) a blank string, in which case the server is supposed to choose whichever style it wants. Since the capabilities document included information about the styles, we are using option (a) and passing the server the name of the style it said it supports. QGIS and other clients don't bother specifying the styles and are using option (b). The server should support both (a) and (b), but it only supports (b) and fails with (a). That's what the issue is.
Here's the full text of the relevant section in the spec:
The mandatory STYLES parameter lists the style in which each layer is to be rendered. The value of the STYLES parameter is a comma-separated list of one or more valid style names. There is a one-to-one correspondence between the values in the LAYERS parameter and the values in the STYLES parameter. Each map in the list of LAYERS is drawn using the corresponding style in the same position in the list of STYLES. Each style Name shall be one that was defined in a <Style><Name> element that is either directly contained within, or inherited by, the associated <Layer> element in service metadata. (In other words, the client may not request a Layer in a Style that was only defined for a different Layer.) A server shall throw a service exception (code = StyleNotDefined) if an unadvertised Style is requested. A client may request the default Style using a null value (as in “STYLES=”). If several layers are requested with a mixture of named and default styles, the STYLES parameter shall include null values between commas (as in “STYLES=style1,,style2,,”). If all layers are to be shown using the default style, either the form “STYLES=” or “STYLES=,,,” is valid.
If the server advertises several styles for a layer, and the client sends a request for the default style, the choice of which style to use as default is at the discretion of the server. The ordering of styles in the service metadata does not indicate which is the default.
So, again, the client has two options - it may either specify the name of the style it wants to use, or it may specify a blank string and let the server use whatever it wants. The choice belongs to the client, the server has to support both.
Here's the request that we send (I added a new line after each parameter for readability, you can get the exact URL used by right-clicking the URL and invoking 'Copy link', or just by clicking the URL to open it):
http://gaservices.ga.gov.au/site_13/services/Geophysical_Grids/MapServer/WmsServer? VERSION=1.3.0 &REQUEST=GetMap &LAYERS=Radiometrics2015_ternary_KThU &STYLES=Radiometrics2015_ternary_KThU &CRS=CRS%3a84 &BBOX=112.9205000000000041,-43.7404999999999973,153.6394999999999982,-9.0675000000000026 &WIDTH=965 &HEIGHT=822 &FORMAT=image%2fpng &TRANSPARENT=TRUE &BGCOLOR=0xFFFFFF
This fails and returns text/xml with the following:
<ServiceException code="StyleNotDefined"> Style 'Radiometrics2015_ternary_KThU' not assigned for layer 'Radiometrics2015_ternary_KThU'. </ServiceException>
The style *is* defined in the capabilities document, but the server says it is not defined.
Here's the request that QGIS sends:
http://gaservices.ga.gov.au/site_13/services/Geophysical_Grids/MapServer/WmsServer? VERSION=1.3.0 &REQUEST=GetMap &LAYERS=Radiometrics2015_ternary_KThU &STYLES= &CRS=CRS%3a84 &BBOX=112.9205000000000041,-43.7404999999999973,153.6394999999999982,-9.0675000000000026 &WIDTH=965 &HEIGHT=822 &FORMAT=image%2fpng &TRANSPARENT=TRUE &BGCOLOR=0xFFFFFF
This succeeds and returns an image, because STYLES is set to a blank string.
The server has to either stop advertising styles in the capabilities document - this is permitted by WGS spec 22.214.171.124.5 for layers which only support a single style, or to recognize the name of the style in the STYLES parameter for GetMap.
Hope this helps.