If all that matters in forecasting is getting an accurate forecast then why is using non stationary data a problem. Say you use non stationary data to create a forecast that performs well in a pseudo out of sample forecasting test would it still be wrong to have it as your model?
Problem is that in case the data are non-stationary your out of sample forecasts will usually be poor.
Even if it by random chance performs well in your training out of sample set, it is very likely that was just some good luck.
If you find that such model consistently outperforms models where you correct for non-stationarity you could use it. As you say in forecasting you only care about accuracy, it does not matter whether you get the accurate results from rigorous model or read them from dried up chicken bones.
This being said I would be extremely surprised if you would build forecasting model with non-stationary data that would consistently outperformed models where you correct for it.
PS: I assume above we are talking about models that cannot handle non-stationarity. Of course if we talk about some error correction model that can handle non-stationary series then there is no issue.