Okay, so control access to the platform. That may be easy enough at St. Pancras, but not so easy at Farringdon where there are multiple entrances. Furthermore, if you start controlling access to the southbound Thameslink platform, you've now created a problem on the adjacent Met platform. The people still have to go somewhere, they don't just disappear into thin air! And whilst this is going on, more trainloads are arriving every couple of minutes on the interchanging lines.
Yet the incident still happened. As I said before, with the member of staff standing just metres away, and unaware of the unfolding incident until it was too late. The incident was only picked up because the driver happened to observe something of concern on his in-cab monitor after the train had started moving. How will your whizzo dispatch staff stop a train if something happens at this point? LU's monitors continue broadcasting to the driver until the full length of the train has left the platform of course.
I think you'll find it would need to be a bit more than 30s or 1 minute to be effective. Indeed, you can end up with the entire station in chaos, and possibly the surrounding streets too. Ever used Holborn in the peak hours? And, of course, with many of the Thameslink services being half-hourly, miss the train and it's a half-hour delay. In that situation I'd certainly be making sure I maintain my spot on the platform to ensure I get a seat when the train eventually does arrive.
LU's crowd control is a little different because generally their platforms clear very quickly, mainly because there's less variation between destinations. Only the Northern Line has a divide very close to London, and even then you're still talking about a train every 4-6 minutes to each route. Thameslink is a completely different ball game - at least five different service groups heading north, much less frequent than LU. And many more people travelling further and therefore keen to secure a seat.