Actually Green Park has had a lot of money spent on it in recent times. For the 2012 Olympics it was identified as a key central London point to make step free, and a considerable amount (I heard over £100m) was spent installing lifts, as much to give step free connections between the three lines as to the surface. The result does however seem something of a route march for such passengers making for the exit.
There was a further major project to make a new and substantial second exit on the south side of Piccadilly, which has actually impinged into the park itself, and has also given a direct entry into the park. It has though certainly improved things at the old main street entrance over the road, which is nothing more than a set of steps in a corner of the Marks & Spencer shop.
Of course, being done by a modern architect they just had to make the new south side entrance building completely out of sympathy with all the classic buildings around - the nadir being the long metal canopy over the new entrance, which looks like a piece of old tin - although I'm sure it was very expensive.