It is appropriate to use an improper scoring rule when the purpose is actually forecasting, but not inference. I don't really care whether another forecaster is cheating or not when I am the one who is going to be doing the forecast.
Proper scoring rules ensure that during estimation process the model approaches the true data generating process. This doesn't sound bad because as we approach the true DGP we will be also doing good in terms of forecasting under any loss function. The catch is that most of the time (actually in reality almost always) our model search space doesn't contain the true DGP. We end up approximating the true DGP with some functional form that we propose.
In this more realistic setting, if our forecasting task is easier than to figure out the entire density of the true DGP we may actually do better. This is especially true for classification. For example the true DGP can be very complex but the classification task can be very easy.
Yaroslav Bulatov provided the following example in his blog:
As you can see below the true density is wiggly but it is very easy to build a classifier to separate data generated by this into two classes. Simply if $x \ge 0$ output class 1, and if $x < 0$ output class 2.
Instead of matching the exact density above we propose the below crude model, which is quite far from the true DGP. However it does perfect classification. This is found by using hinge loss, which is not proper.
On the other hand if you decide to find the true DGP with log-loss (which is proper) then you start fitting some functionals, as you don't know what the exact functional form you need a priori. But as you try harder and harder to match it, you start misclassifying things.
Note that in both cases we used the same functional forms. In the improper loss case it degenerated into a step function which in turn did perfect classification. In the proper case it went berserk trying to satisfy every region of the density.
Basically we don't always need to achieve the true model to have accurate forecasts. Or sometimes we don't need to do really good on the entire domain of the density, but be very good only on certain parts of it.