For a live game, I wouldn't want this, since the memory element is a portion of the fun of the game.
But that element of the game design disappears of course in a contest that plays out over a long time, and to the detriment of the game.
You would need to implement it so that the number of your non-trackable trains (your starting trains) remains "hidden" (in silhouette or whatever).
Have lost a couple players to this issue, who thought it was "not fun" that you end up kind of just playing blindly (and nobody cares enough about a light family game like Paris Connection to take the time for notes, or log-scrolling).
This should of course be optional, and not the default, since many won't care for this idea.