To be honest, I can very much see the point of this at Paddington. Great Western services have been in a rotten state for a few months, thanks to a series of infrastructure problems and the well-documented IET availability issues. Services are being chopped, replanned, turned back short on a daily basis.
The first time I heard the announcement was yesterday. It was absolutely the right call. Services were leaving out of order, as and when drivers were available and delayed inbound trains turned up. The announcers were doing a good job of telling people which would be the next service for Reading, how long delays were going to be, and so on. You wouldn't have got that from the app (and yes, I was looking at RTT as ever!).
When the railway is running better, and it absolutely should be, then yes - information should be seamless and it shouldn't matter whether it's delivered to the passenger via a tannoy announcement, a platform screen or a phone. But in the situation as it stands right now at Paddington, I can absolutely see why they're making these announcements.