the law does not agree with you. an 11 year old is still a minor. Lounge entry or fare rules are independent of that law. (and in the same way, an airline charges a passenger over 12 the full adult fare does not mean that child has the right to sit at an exit row 'as an adult'). If the lounge staff approved the 11 year old to sit there a liability potentially accrues.
Once again, your assumption that a child will take less time in their own shower is simply that (an assumption). There is no proof except for your family
Our family is different. We don't shower together (at least not at 11 - that would be a bit strange to us). Time taken in the shower varies according to lots of things. If the 11 year old had long hair and decided to wash and blow dry that then they could take longer.
I'm not talking about the law, wrt the treatment by the lounge staff. As you say lounge entry is independent of the law. The lounge staff don't have to approve anything, They have already stated that person cannot enter. What do people who are waiting for someone in the lounge do? They sit in the waiting area like anyone else refused entry. Now if that independent law about minors creates a liability for BA, that's entirely their choice. Just as the parents can make their own choices about what they allow their child to do.
Or going back to my previous statement if the lounge staff treat the child as an adult under their lounge access rules, that are independent of the law, then it is good enough for the child to sit in the waiting area as a consequence of those independent lounge access rules.
I think you'll find that being a minor is only coincidentally related to children not getting an exit row.
The shower time is not an assumption it is simple logistics. Adult showers, then gets out, dries, gets dressed, dries hair, applies makeup, cleans teeth, repacks bag. While the adult is doing all those things after getting out of the shower, the children can shower, get out, get dried, get dressed. There is an overlap in the tasks such that they can be performed concurrently. This results in efficiency in using the facilities. It also means that only one shower "cubicle" is occupied not 2 or 3, further efficiency. The required time is naturally reduced by these efficiencies.
As for the showing together. 1. Same gender together, I don't shower with my children. 2. We only do this in lounges when efficient and quick use is required because there are other people waiting who might have a tight connection. 3. A full wash, condition and blow-dry of long hair is not undertaken for the same reasons as 2. 4. What we would normally do at home doesn't apply to a lounge out of consideration of the needs of others. eg I don't read a good book for 30 minutes before having my shower.
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