Why would the drivers have to do it?
Because the PIS is built that way. The Driver is given a code on their diagram and then input it to the onboard PIS machine.
On SWR it seems to be run off GPS (which admittedly occasionally leads to odd results when the train thinks it's inbound (on the up Windsor) when it's actually outbound (on the down Main) or vice versa).
The SE version also runs off the GPS.
Far from controlling it, the train crew are evidently unaware of the PIS's behaviour.
What you see is the result of the PIS changing. What also happens is that when the PIS changes. Passengers pull the passcom because they think the train has been diverted. You also get passengers coming up to the cab window asking where the train is going because they believe the train is terminating or they aren't sure if they are on the right train. When the PIS doesn't match where the train is going it causes problems.
The onboard PIS works the best when it shows a single direction. It is the best reflection of what the passengers are actually doing. When you get to Slade Green you need the PIS to state the train is going to Cannon Or Charing Cross. At Slade Green the service becomes a Slade Green to Cannon/Cross so the most honest and realistic PIS display will be whatever the stopping pattern is with the train terminating at Cross or Cannon. SE Metro tend to run into or out of London. Even the rounders are still Up+Down. The aren't thought of as 'circulars'
I used to drive the Sutton Loops down from St Albans and into Blackfriars. Keeping it simple as 'Via Wimbledon' is all that's needed because it reflects passenger behaviour. Same with anything going through 'The Core'. There is a clear split with passengers going into London and those going outwards. The trains empty out in the middle bit but have a small overlap in the core. Passengers see where its going 'Via' and get that train accordingly. The loop ends or only lightly used and tend to be used by locals who know where they are going.
Via Sidcup, Via Bexley'eaf, or Via Woolwich is all that's needed as it clearly defines the direction of travel. Flip it over at the bottom end of the loop and have it reflect that 'This is the service to London Cannon/Cross' because that best represents the passenger and the service. The CIS only needs to avoid a situation where you have the up and down platform both showing as Cannon/Cross.
Personally I don't care what the CIS says as I rarely trust it and I tend to know what train I need before I get to the station. (just like the majority of commuters) What I do care about is that when I'm on the train, it's telling me what the train is doing and not giving me information I need some kind of inside knowledge to work out.
When I'm on a train as a passenger I love to people watch. People on the train rely heavily on what the PIS is saying. When someone wants to know where the train is going, they look to the onboard information.
I freely accept the Tube method and as I already mentioned, you are pretty much always going a few stops. You only care about direction and its backed up by the intuitive mapping system. I get to a tube station check where I'm going (line name) and then see if I need north/south/east/west. The entire system seems to be geared to what the passenger needs. I'm going 4/5 stops at best and I only need local information. I couldn't tell you where my tube service goes. Only that I get to Embankment, check the map and go left or right down the stairs to the platform. I have zero idea where my train is going. I get the next one that arrives, go a few stops and then get off. Trains are different as they lean towards an end to end mentality. I go towards or away from London with a huge emphasis on 'Via' direction of travel. Towards London is going to work; away from London is going home.
I've always found SE Metro to be awful for locals. I've lived in Kent for 30yrs and local travel via train is crap. Trains go into or out of London so nothing goes across the County. Going from Sidcup to Bexleyheath is far quicker and more convenient by bus/car than train. Orpington to Swanley is a Drama Lama and going to any of the town centres is better by car/bus because the stations are so far away from their town centres. People who are going across the county are taking the bus. Passengers aren't taking the trains round the loops. It's an infinitesimal number of passengers going round the loops. At best its passengers going 1/2 stops and as
@MikeWh stated there are The Charlton home games to consider.