Entropy during neuronal signaling
In my view, thinking of entropy as something biological life has "wants" about is a bit mistaken. In general, entropy arguments regarding biology run into a major problem in that the interesting physical laws involving entropy required closed systems, and no biological system is closed, so it's a bit moot.

It makes more sense to think about what benefits "fighting against entropy" gives a biological organism, and I also think it makes more sense to think in terms of potential energy rather than entropy (even though they are of course related).

Neuronal signaling is best when it's rapid. By pumping ions against their gradient ahead of time, neurons establish a concentration gradient (which you can view as a decrease in entropy). This is indeed the active part. This also stores energy in the gradient, in that if a 'gate' were opened, the gradient would tend to equilibrate. Because the ions involved are charged, additional sensors in the membrane can respond to this changing electrical gradient.


  I would have hence found it more intuitive if signalling would be acquired by an active decrease in entropy (followed by some sort of "passive" restoration)


I think you might be mixing up the "action" verbs here. Think about pushing a rock up a cliff or lighting a firework. If you want that rock to be a signal, it's going to be a much more dramatic signal falling down the cliff than it was pushing it up the cliff. It's going to be much more dramatic lighting off the firework rather than extracting the needed ingredients from various places and assembling them. Processes that release stored energy tend to be a lot more dramatic and "active" despite being "passive" in terms of needing to add energy (besides activation).