All infrastructure is a matter of trade-offs. Sure, they could have built the platforms 3 times wider, with funnels leading to the doors, and gates that open/close to prevent people from falling onto the tracks. But that would be an extreme increase in cost to build, and wouldn't lead to a significantly larger number of people able to take the train. This app doesn't improve matters: it causes an additional "tragedy of the commons" situation. By being the first one on the platform, you may get to claim your seat, at the common cost of making it harder for everyone else to get out of/on to the train.
Yes, at peak times there is congestion (just like at peak times there's highway congestion), and the existing infrastructure (built mostly before 1930) was built to be significantly safer and more efficient than existing systems. And 90 years later, it still is usually the most efficient and safest mode of transport in NYC. But given the record number of subway riders and record age of the infrastructure, it takes everyone pitching in a little to not make the situation intolerable. Part of that "pitching in" is "Don't go onto the platform before the train's riders have left".
Except that (correct me if I'm wrong), the platform is displayed 10 minutes before the train arrives, so there is already people waiting when the train arrives
In my experience after commuting for a few months, this is never the case. Even when the track listing is displayed 10 minutes prior to departure time, the train usually won't arrive for another few minutes. You get crowds of people where the doors will be. It's really not safe, although I've never seen someone fall so it can't be that bad.
Yes, at peak times there is congestion (just like at peak times there's highway congestion), and the existing infrastructure (built mostly before 1930) was built to be significantly safer and more efficient than existing systems. And 90 years later, it still is usually the most efficient and safest mode of transport in NYC. But given the record number of subway riders and record age of the infrastructure, it takes everyone pitching in a little to not make the situation intolerable. Part of that "pitching in" is "Don't go onto the platform before the train's riders have left".