Yet again, the point appears to be missed - AA did not issue a boarding pass to Italy. They issued a boarding pass to LHR. If the passenger turns up in Italy, with no onward ticket, having been transported to Italy from LHR by BA, what has that go to do with AA? BA may have a problem but why would AA be at fault? The issue is not one of "personal responsibility" - the issue is one of AA making up non-existent rules such as "you need a return ticket to the US to travel internationally" - complete and utter rubbish.
You are exactly correct - it would be BA's problem.