First part of the TLogSysLogProvider.WriteLog method:
procedure TLogSysLogProvider.WriteLog(cLogItem : TLogItem);
var
msg : TIdSysLogMessage;
begin
if not fSysLog.Connected then fSysLog.Connect;
msg := TIdSysLogMessage.Create(nil);
Regardless of Connected returning False, Connect causes an error on iOS. Since UDP is connectionless anyway, I fail to see why TIdSysLog even has the method, or why TLogSysLogProvider uses it.