In addition to this, the ?WSDL link was not being shown either.
Eventually I tracked this issue down and it had everything to do with my packaging. My EAR layout was like this:
and the WAR layout was like this:
Basically the EAR file contained all the additional JARs and the WAR file had just the code for the web service itself. Since the WAR file is a module of the EAR/Application, when looking at the Testing tab of this module I'm guessing the WebLogic Console had issues resolving the necessary classes loaded at the application level. The result was that the service was reported as unavailable incorrectly.
The fix for this was to move all the 3rd party JARs into the WAR file WEB-INF/lib directory instead.
The EAR file layout was now like this:
The WAR file layout was not like this:
Once this was redeployed, the WebLogic Console was showing all the correct information and the ?WSDL link returned.