In a perfect world yes it would. Such a system would essentially rely on dispatch staff/guards/drivers having access to real time information about connecting services being delayed, covering every location (or at least access to a feed from the system processing the information and making the decision for them).
The system processing it based on quite detailed criteria (including stuff like through Advances sold and e-ticket scans) is what I'd envisage, FWIW. It could take other things into account, such as what the connecting train is doing. If you consider my theoretical Cornish branch example, if you imagine that the inbound from Paddington is say 15 late, but it is already known that the fault on the train will mean the next Paddington connection is definitely cancelled, then the overall benefit may be best if the branch train waits and the following round trip is cancelled to bring it back on time, because there's already known to be nothing for it to connect with, and the people in whatever charming little coastal town it is might be better served by spending an extra hour in the pub, on the beach, having chips or whatever than standing in the cold at an interchange station with few facilities as many of the Cornish ones do.
It’s fairly clear that such a system doesn’t exist, and the funds to provide it won’t be forthcoming. Therefore the only sensible real world solution is to prioritise on time departures for “the greater good”, with a few limited exceptions. Ultimately by doing this you’re reducing the number of missed connections that will arise in the first place, which is surely the objective of the exercise.
I'd go with what
@Watershed says, to be honest, that is that each station should have a "risk assessment" done for each connection (clockface makes it easier, as it'll be the same all day) to establish how long it should be held and in what circumstances, so passengers on incoming trains are informed in advance of what will happen. This would include such things as "if it's -5 outside and the station has no shelter, it is unacceptable under any circumstances to strand passengers there for an hour", which might include such things as pre-arranging taxis or communicating to the train that passengers should take some other action, e.g. continue to Scarborough where waiting facilities are available.
It's not
simple, but it's worth doing if you care about customer service.
But looking at the example given, I would say under say 3 minutes = always hold unless there are specific circumstances. Closing doors in peoples' faces as they cross from a train that's already there is unacceptable, and a train under 5 minutes late is in some ways not considered late at all. Some guards might have held it anyway and told a fib about assisting a disabled passenger when asked "please explain", but they shouldn't need to.