I agree that things should be sharpened up a bit.
Apart from no presentation of passport required to cross to airside and reach the gate at one major UK airport we also have a new routine at the same airport which involves cursory pre-inspection of passports 'in the vicinity' of the gate. In an effort to trim boarding times even further the inspections now often do not occur 'at the counter' but they now come to you McDonalds Pre-Order style in the (open) queues, with no further presentation of anything required to walk through to the aircraft ten minutes later.
Very slack.
As for captain in charge bit, the law says it's from when the doors shut I think, which is perhaps why we get all these exploitative variations creeping in.
And as for who is potentially liable during boarding for any injuries well I guess with some airlines it depends on whether you board at the front or the back.