I find it unlikely that the person who boarded with the incorrect BP had been rebooked on the OP's specific flight. Had that been the case, he would have been brought the correct BP and moved seat, unless a further mistake was made.

By 'good to travel' I would infer that he was admissible to the UK (hard to obtain the KL BP otherwise, assuming genuine and belonging to the passenger), had been rebooked on either another BA flight or a KL service, had presented at the gate with a passport in a non-Western name and a BP was reprinted for a different non-Western name by a staff member not paying attention.

The procedures as written by and large do a good job of checking documents, but it is impossible to ensure that all the frontline staff really understand what they're doing and why.

Case in point: at LCY recently I showed ID on boarding and moved directly to scan mobile BP. I didn't pause long for the staff member to read the text on the mobile BP, and as it was face-down on the scanner he made a point of telling me how he would need to see it afterwards to ensure the name matched my ID. He didn't look at his screen or the BP reader, maintaining eye contact with me and then checking mobile BP against ID.

In my opinion, as the text on a mobile BP essentially means nothing as it can be modified with Microsoft Paint, the proper procedure should be to match the ID to the name displayed on the BP reader. Otherwise, Mr Inadmissible could check in online using name and passport details belonging to Mr Citizen, the details having been easily enough obtained from a hotel front desk employee, with Mr Citizen none the wiser because he still has his passport. Mr Inadmissible edits the self-print or mobile BP to show his own name, and then turns up at the gate with entirely genuine ID matching the BP text. That is presumably why the BP scanner displays the name from the airline's records.