SIP prevents Lync 2011 from starting
Installing the "OS X El Capitan 10.11.6" Update fixed the issue. This is perhaps not a proper answer, since I still don't know the underlying cause, but I post it anyway since it points to the cause being configuration of OSX and not a problem with Lync. Since it was fixed by the upgrade, I most probably could have configured my previous version of OSX (10.11.5) to fix it but I don't know how.