The only real thing that I see that is missing from suggestions on here (Unless I have misread or skipped something) is that the ideas presented offer no time pressure training.
There are two main ways to count a game… and both are valuable in different stages of the game. There’s the educated guess and the actual count.
But there is something that links them… time.
In different parts of the game, it is either easier or more difficult to count the Game. After the fist fifty moves in a game, the count is an educated guess over which slices of the pie are going to get points. But it’s just a guess. But later in the game, there is a more certain count as there are fewer variables to work out, and the count becomes more accurate naturally. But time pressure is something that can really skew how well a person is able to count.
So my suggestion would be this… Three or more stages. Each with time controls.
Beginner: over a min to count (The board is a completed game)
Intermediate: thirty seconds to count (most endgame is incomplete)
Advanced: between 15 seconds and thirty seconds depending on the board. (The board can be any of the above, and indeed should include games still in the midgame. Because knowing if a player is ahead r behind can change the type of moves that a player has to play to either secure the lead or to gain the lead from the opponent.)
Of course, there could be alterations to my idea here, as I have never developed anything like this in my life… and this is just an idea. But I think adding time pressure could very well be a great way of actually training people to count in a more realistic way.
In terms of the randomly generated board… I don’t think it should be all that random… There is already a database of all the games ever played on OGS. Some games are completed games, some are not. My point is that we already have the resources to support such a thing being created. And the boards need not be all that well played out. As in… a low kyu game could be as valuable as a high dan game, for counting, because of the different situational problems around the board.
There is also the added issue of prisoners. This is something that should be factored into the counting as well I think. So some boards have captured stones to be added to the actual count of the board. This adds in a complication that all players should already be fairly familiar with, and it’s realistic to have this in there as well.