I agree with the previous poster that this wasn't a rescue service, but because you weren't rescuing the passenger - he was able to complete his journey using his booked operator, on a later service. All you were doing was mitigating his delay.
Just because alternative transport (be that another train operator or a taxi) could get you to your destination faster, does that mean you are entitled to alternative transport?
Yes, if the delay in your arrival time at your final destination is expected to be over 60 minutes.
And at what length delay would that entitlement kick in: 2 hours, 1 hour, 15 minutes?
As above, 60 minutes. It's retained EU law - it's not optional.
I wish you were entitled to alternative transport if it got you there faster. For many of my journeys, a taxi would always be faster than train.
I don't think it's necessary for transport operators to organise taxis in order to comply with PRO, if trains are still running and the expected delay is within the bounds of what is reasonable. I don't think the concept of "rerouting at the earliest opportunity" is as mysterious as some like to make out, and if you're trying to argue that a transport operator has not re-routed you at the earliest opportunity, plainly you have a stronger case if a scheduled train ran and could have got you to your destination quicker, than if no scheduled train ran but you contend that the TOC could and should have sorted out a taxi. Which is not to say you have no case in the latter scenario, merely that it's a more difficult case to make.
As for why people expect to be carried on another train, I should have thought this would be obvious but it obviously isn't, so here's the reason as I see it. People typically understand that if they buy a ticket for a specific train and it says "Booked Train Only" on it, then that is the train they have to use in normal circumstances. It is also pretty obvious to all concerned that if it turns out to be impossible to use the booked train due to a failing on the railway's side, the passenger is still entitled to get to their destination, which inevitably means using another train if one is available.
What people don't necessarily understand - and there's no reason to expect that they should or will - is that, layered on top of the "Booked Train Only" restriction, is a TOC restriction, and the TOC restriction may still be enforced even where the Booked Train Only restriction cannot be. That rule is not obvious to non-railway people. It exists only due to the vagaries of revenue apportionment within the industry. People don't get that. They shouldn't have to worry about that. The industry should sort that out itself without troubling the passenger.
It's heartening to learn that descretion was exercised sensibly in this case, but I don't think passengers should be reliant on descretion in such cases.
It's an interesting thought experiment, actually, to consider if the passenger's position will be any better once GBR has taken over all the passenger operations except open access? The current scenario is a person booked to travel with Grand Central, with XC as the connecting leg, ending up on an LNER train and dealing with an LNER train manager who doesn't work for the company that caused the delay. Presumably in an exactly comparable situation once GBR has taken over both XC's and LNER's operations, the train manager would not for one moment suggest that the passenger should wait for the next Grand Central service to London, as they would recognise that GBR had caused the missed connection and should mitigate the delay by conveying the passenger to London as quickly as possible? Or is that too optimistic?
