just correct something, Long memory models in fact incorporate both stationary and nonstationary process that is , when d=0 or d=1. Depend on the values you estimate.
Again, long memory is not just for long time dependent. In fact, when you have mixed data like mixed unit root and stationary data, long memory will play an important role in it. For example, monthly inflation data,output growth data, daily stock index data exchange rate data. Sometimes, even for most unit root test, it may misspecify the long memory process as unit root process.
Well, first, do not forget about low power and size distortion among unit roots tests.
Second, the purpose of models with time series analysis, especially in ad hoc method, is mainly on forecast.
If a model or some econometric skill neither incorporate theories well nor forecast reasonably and acceptably, why do we need pay "more" attention on them? Besides, computation cost is always the first in practical.