The concern was related to being able to know where emergency vehicles were. If you build a system that announces to traffic light “I’m police/fire/EMS, coming through”, you also build an early warning system for criminals and terrorists who either want to avoid or target you.
I think these systems used RF before, so anything in the vicinity could listen in. That said, if this new system is a phone app, it'll just be using phone data instead, so it'd be hard for that to be intercepted.
If we are assuming terrorists that can get real time access to government servers, we should assume they will access the servers used by the dispatchers and so can find out when an emergency vehicle leaves and where its destination is.
That is probably more useful for planning an attack on emergency vehicles.
Although why even bother with servers? Just call in a fake emergency (or cause a real emergency) at a place of your choosing, and the government will send emergency vehicles to you.
At least in the UK ambulance crews are extremely tightly monitored (mostly for good reasons) so surprised there would be any privacy concerns for crew on the clock.