Providers like Timatic and Traveldoc are a source of information for travel requirements, nothing more. Carriers can and often do have other means of verifying a passenger's travel requirements in case of a dispute, if they choose to do so.

Barring some direct instruction from the relevant authorities (which clearly isn't the case here), the decision to board a passenger is solely the carrier's to make. They could've escalated the matter to a liaison team, but they didn't. Whether it was because they ran out of time or willpower, I don't know. But the point remains: it was KLM's call to deny boarding, not Traveldoc.