For a claim to be approved evidence is usually presented, evidence checked and if correct the claim is paid out.
I'm not defending the letter sent out by LNER in this case, but...
The fact is, it's not very difficult - for anyone so inclined - to defraud the Delay Repay system, if they have any kind of flexible ticket (it doesn't even need to be an anytime).
If the ticket is valid on a number of different services (whether on one day or over multiple days) it's now quite easy to check retrospectively whether any of those services suffered a serious delay. And then, when the delay repay system asks you for the intended start and finish times of your journey, it would be easy to enter times that matched the delayed service rather than the one you actually travelled on.
I have never been asked to submit any kind of evidence other than the tickets themselves (and/or booking receipts). And in many cases, as far as I know, there's no way to tell from that "evidence" which services I actually travelled on. Sometimes a paper ticket might be stamped with a train headcode, but that seems to happen increasingly rarely. Often they'll just have a scribble on them. Sometimes they simply won't have been checked anywhere on the journey. Maybe they will have gone through some gateline machines but as far a I know these don't record or communicate anything about a specific ticket to anywhere (maybe I'm wrong).
Maybe I've never claimed for anything controversial enough but I've never been asked to submit any other form of evidence, anything beyond the tickets themselves, to demonstrate I was in a certain place at a certain time.
99% of the time, if a claim is challenged or if they ask for additional info, it's because of some kind of bureaucratic silliness or error at their end.
(To be clear, I don't engage in this kind of fraud myself, but I have always been aware of the possibility, and slightly surprised that such a large loophole is left open)