Must admit I like the "silent airport" approach of only announcing really important things but having screens everywhere. Surely there is a better technology based option for serving blind and partially sighted people the information they require directly than continuous droning?
Since the purpose of both the screens and loudspeakers is to provide information that passengers will find useful, voice announcements have the advantage that you can hear them (almost) wherever you are on a station, and they can attract your attention when it's on something else. But you have to be looking at a screen to be aware of what it's displaying, and if you're doing something else you have to remember to switch your attention from that to a screen in order to get the information you may want.
Your point about "continuous droning" illustrates the downside of loudspeakers: all kinds of other stuff can easily be added which is of limited interest and unfortunately has the effect of causing people to pay less attention and thus miss something they wanted.
I'm sure the technology could easily be developed to insert a chip into the ear of people who are blind or have seriously impaired vision, which would pick up a digital signal and convert it to sound in order to let them know that their train was approaching (or had been cancelled for the next two months, in the case of certain TOCs). [I can't find an icon for "I didn't really mean that last bit".]