Over the years they have been tweaking the level of detail and presentation of status displayed to the train Op and no doubt this will be reviewed again under the CLIP work.
Yes the view/replay of the fuller scenario/all events (typically called the Train download) associated with a reported failure is very very useful in accurate diagnosis/repair.
On the current 92 stock the data is only "uploaded" for remote viewing when train is dormant (eg stabled), I think the refurbished trains will upload much more regularly (as per the S stock, where I think the viewing facility is available in the control center as well as in maintenance depots)
With the leaps in data processing technology, Remote condition monitoring has been a growth area (in many industries including TFL) and enables monitoring/scenario searching of the data for trends by train and/or location highlighting deterioration for early attention. Currently useful for spotting units with failed WSP controllers, flashed over motors, Speed sensor faults, Doors detection, loss of Auxillaries, ATP/ATO controller issues, ATO Platform stopping accuracy/train braking performance, weak Trackside signalling signal levels to train.