What WS Addressing does is allow us put the URL target in the actual SOAP Message - correct? Hence we don't rely on HTTP for this anymore.
Correct. Although HTTP -not knowing anything about its SOAP payload- still needs the URL as well, like it does for all requests.
But, can someone give me a example of a realistic business case where using WS-Addressing would make sense to use? And would not make sense i.e. would be overkill?
WS-A isn't something that a developer would generally use directly, but several other WS-* standards rely on it, notably WS-Security. A WSS message might contain something like this:
So in addition to the service URL (which some part of the SOAP processing pipeline might use), there's also a unique message identifier (important for WSS) and the SOAPAction attribute (which -being part of the SOAP header instead of a HTTP header like it used to be- is now also available to the full SOAP pipeline).