That appropriate arrangement is often dropping off/collecting from stations... Are you saying children/any other vulnerable person shouldn't be allowed to use the railway unaccompanied to get to school/relatives?
It is unwise for a vulnerable person to be using the railway without putting some mitigation measures in place in line with their vulnerability. When I was a kid it was that I knew my parents' BT Chargecard number off by heart, so even if I had lost everything I could make a call home, though these days it's probably more likely a mobile of some sort. The railway doesn't provide an "unaccompanied minors" service - that doesn't mean unaccompanied minors can't use it, they can of course and very often do - what it means is that the parents remain responsible for them and, other than minor things like not chucking them off in the middle of nowhere if they've fare dodged, don't really have a duty of care above that to an adult.
Other wise options are things like 20 quid hidden about their person for a taxi if needs be, or a GoHenry type card loaded with money for that situation.
Having said that, this situation is an example of terrible incident management.
As soon as Control knew that train would terminate there, they should have:
1. Arranged what was going to happen, be that a stop order, a RRB, taxis or whatever. Or if they couldn't quite arrange it yet, bear this in mind.
2. Communicated it to the guard of the terminating train so they could announce it (even if it was "we know about it, please wait on the station and an update will be announced by time X". If the situation hadn't changed by time X, *still announce* even if to say that the situation hasn't changed*)
3. Announced it periodically over the PA
If that wasn't going to be viable, they need more staff. Yes, I know "aviate, navigate, communicate" applies, as it were, but that little phrase doesn't mean you don't communicate, it just means you have to do the operational stuff first or the job stops.
My experience is that railway incident management is very poor and would do well to have some ITIL** type principles applied to it.
* This is one thing airports do reasonably well - they post "more information at <time>" on the board if there's a delay, and at that time do an announcement even if it's to say "no idea yet, we'll update again at time Y".
** IT Infrastructure Library - in practice this is stuff like incident management.