There is no magic threshold of 'Public Service'.
I agree there is no 'magic threshold' but my view is that once subsidies are in 50% area it is not longer a viable business. My arguement to support that:
At 10% subsidy, if it was removed, there would be some economies, rural services may be thinned, but the operation could continue pretty much as it is now with good management.
At 25% subsidy, then removed there would be some tough decisions, but I think it would survive as an industry/network.
At the current level of subsidy, 50%+- if that was removed we would be looking at Serpell/Beeching Mk2 and the resulting business would look nothing like the current operation.
I am not sure if the currently profitable lines would remain profitable if the connecting feeders were removed, if people had drive further would they just do the whole journey by car for example? So in that scenario, where a future government pulls the plug (unlikely, I know) the current network is finished.
To me that meets one of the criteria for a 'public service'. Under the current system all capital expenditure has to be signed off by the DfT, thats another indicator for a 'public service'. All fare box revenue goes to DfT, another indicator, so to use the old phrase, "If it quacks like a duck and walks like a duck it is a duck"
If I buy a ticket from A to B bought at a station owned by train company Z then I want to travel on the next train
I think the whole ToC thing with respect to fares, ticketing and which trains you can use needs to be binned, its an additional complexity which doesn't not benefit the passenger in any way and causes friction with front line rail staff when people are caught out by the restrictions in some way or other. If you want encourage people to use less busy slower services for example then do that by offering competetive advance tickets which can be bought right up to time of departure, but are tied to that train. (Northern do this anyway on quite a few routes)
it adds 2 to 4 minutes to change at Leeds onto an LNER service
But people dont want to change... Passengers have learnt that changes increase greatly the risk of a disrupted journey, because in general we do not hold connections, and any journey involving more than one ToC comes with added riskes of complications. The less mobile also value through trains, as do those who have luggage and/or children. And a through journey is usually more pleasant.
Take Hull to London. assuming I have some degree of time flexibility I can get on a Hull Trains service in Hull and be taken all the way to Kings Cross. Because I get on at the start if I have a reserved seat I (probably) wont have to tip someone out of it. If unreserved I will be able to find an unused seat that suits me. The alternative, Northern to Doncaster, to change on to LNER. Journey time is a bit longer but at Doncaster I will joining an overcrowded LNER service, 50/50 there will be someone sat in my seat, and if there is disruption on the Northern leg you then have to convince a grumpy LNER revenue person that the reason you are on the wrong train when you are travelling on an advance ticket is because Northern delayed you. All additional avoidable stress, so next time you get the through service.