Well that paper thing survives, in a way. You will recall a period when both paper tickets and online bookings co-existed. This was only possible because a reservation still had a ticket in it, be it paper or electronic, and some aspects of ticket processing from that era survive to this day. So a PNR is a wrapper around a reservation for a flight, for hotels, maybe excess baggage. All exist electronically. Payment is via the e-ticket, each PNR can have one or more e-tickets, and this has to be attached to the reservation for the airline to be able to drag money into Accounts Receivable. Hence OLCI won't work if there is no e-tcket since BA won't know if it will get paid. If a booking is totally touchless and simple, then automatically that happens and I would guess that's over 95% of bookings that are touchless / self serve. But FTers love to do things in special ways, and that often means the e-ticket has some level of manual processing. Had the OP made the booking fully online, or fully in the Contact Centre then almost certainly it would have worked fine and the e-ticket side would be automated. However s/he did a mix, and thus manual intervention on the ticket is needed. That's not the customer's fault, BA should deliver on its own processes - and ideally iron out process failures one by one.

Best practice is to check the BA App to see if you have an e-ticket number in the booking a few days before travel. If you have an email with the e-ticket number, dated after your last change, then that's good enough.