#4 No shadows over him
One of the main reasons for Neymar wanting to leave Barcelona is to get away from the shadow of Messi. While there is no doubt that they share a great understanding, there is also a point that he will never be the main man as long as he plays with Messi.
The same situation applies for Coutinho as well. He will have to not only fight for playing time but also for recognition with the likes of Neymar, Messi and Suarez. Further, if Neymar stays, there will be constant comparison because of their history unless Coutinho changes his position.
However, at Liverpool, he is the no.10 and the main man for Klopp's Reds. There is no shadow of any player over him and he is free to carve his own destiny.
Moreover, there is no huge price tag over him to increase expectations. It would be the case if he moves to Barcelona as Liverpool will not sell for less. The absence of a release clause strengthens the English side's position further.
Thus, it makes more sense for Coutinho to stay at Liverpool and continue to be the talisman instead of having giant shadows over him at Barcelona.