Anything that needs manual ticketing / goes to the back office is prone to get overlooked. There is a set of queues and agents sometimes don't correctly flag items. And what you experienced, in different forms, has happened many times before and will do so again. You are right to check the presence of ticket numbers particularly for non automated ticketing scenarios like this. This is easiest done in the details panel of the BA App where the information is easy to spot and the absence of a long number beginning 125 takes just a second to do. That said it is pretty rare, probably well under 1% of travel, perhaps well below 0.1%. But my experience is that FTers tend to complicate their bookings more than most - I suspect most people would have booked all 3 tickets via the contact centre rather than adding one afterwards.

Airlines do try to shuffle out of the responsibilities here (BA isn't the only airline where this happens) but ultimately you paid the fare and you got a confirmed reservation. So you are able, on behalf of your child, to claim for his denied boarding compensation, as well as the hotel accommodation and what not. Now since he cannot travel alone, I think that probably extends to one adult too, but arguably the other parent could have travelled. Ultimately it was BA that made a mistake here, and componded it by removing staff in LHR who could retrieve the situation.

In the good old days LHR had agents who could push ticketing through manually. Or could in extremis take a payment for a new ticket, issue that ticket, cancel the original ticket and then after travel you could claim the fare differences. However BA took a commercial decision to remove this element of training from agents (and reduce their costs by downgrading the ground staff jobs). So I would not have any hesitation for pinging BA for the denield boarding compensation for all of you, perhaps compromising on kid + 1 adult denied boarding. See the main EC261 thread in the Forum dashboard for this. [If it was an Avios booking then it has long been the case that there are no Avios trained staff in LHR, so that's sliightly different].