Not so few. Apart from peak holiday days, one of the most common causes of gross overcrowding is the previous service being cancelled (or running so late it amounts to the same thing).
How often on normal days do all the southbound services of the day mesh together at say Doncaster in order. In my experience, maybe half the time. What's to be done then? Your booked train from Donny ran into an issue between Aberdeen and Edinburgh, and is two hours late. But here's a service from Leeds running in for London, half full. Are you allowed in or not? And if the Aberdeen originator is now cancelled, what then? A one-hour queue at the sole ticket office window like you get at airports during disruption? If the response is "come back tomorrow" then I can see the police having to be called in short order. What a way to run a customer-facing business.
But in my experience the more likely scenario is it's the train following the cancellation that is packed out. Have people been allowed in? If so in you get, and push through. There's an 80-year old granny sat in your allocated seat. What then?
Silly plan, not thought through with what happens in reality.
