The basic reason for this is really, really simple:
When the method server pings https://<our winchill domain name>/Windchill/servlet/WindchillGW/wt.httpgw.HTTPServer/ping, it cannot successfully ping it, i.e. it does not receive a 200 response code to this URL request prior to the request timing out.
Which method server is doing this ping is largely indeterminant, I believe -- as this responsibility gets handed between method servers. I am pretty sure, however, that only foreground method servers do such pings.
If the foreground method servers cannot successfully do such a ping (e.g. you don't have a web server directly on each of your cluster nodes or you improperly configured to require authentication for this URL), then a failure will always be indicated.
Looking further into the code, "SITE NOT RESPONDING" indicates that other probing requests against https://<our winchill domain name>/Windchill also fail -- leading to the conclusion that the overall site (including simple anonymous static pages usually served by the web server) cannot respond (vs. the method server not being responsive, for instance).