A train dispatcher firstly provides an additional safeguard against a train being involved in a SPAD against a platform starting signal, and secondly provides a pair of eyes to observe the train safely out of the platform. They also are able to provide information to both customers and traincrew. At many busy locations the dispatcher will have radio communication with the controlling signaller, local operations manager and other colleagues around the station, and is involved in co-ordinating movements within the station and managing the platforms. If you take a busy location such as Bristol TM, there is a lot of communication which takes place between platform staff and those running the trains from out of sight, without which things would run far less smoothly. Even in this modern age, not everything happens automatically!
The question "what could they do once a train has started to leave" also ties in with another practise which is sometimes questioned; the Guard remaining at the door position until clear of the platform. Where this is in operation, a dispatcher will generally be positioned further forward than the Guard, and so is able if needed to give the Guard an emergency stop signal as the train moves along the platform. Of course, this is not relevant in lots of places, but it makes good sense where it does happen.
Unfortunately there are some places where the dispatchers do little to help justify their jobs (not generally with FGW I would add, where they are usually to a good standard), taking a very 'laid back' approach and sometimes being selective in which trains they see away, or how long they remain on the platform as the train departs. This is a shame, as it can give the impression that they are superfluous. But, when things are done properly, they are vital to a safe and smooth-running station!