But there is one problem with pure PoS coins:
Rich get richer JUST by sitting on their stake.
which (in my opinion) inevitably leads to centralisation of stake (just getting all fees until you get all the currency).
It will take definitely much longer time than centralisation of mining power did, but it probably will happen some time.
And wide adoption of such currency is only speeding the process (more transactions → more fees)
Am I wrong?
That is the only thing that i dont like about NXT and
that is where NEM with their Proof of Importance algorithm can succeed.
As we know there are two components that add to the inflation: increasing monetary base + increasing velocity of money. Introducing a little inflation in the system helps prevent the issue you just described - people have to transact, not just sit on their stakes, causing deflation in the system.
NXT and NEM fight deflation in different ways.
NXT will have a Monetary System feature in the coming months that allows to run other currencies backed by NXTs.
NXT also allows to trade asset units on the asset exchange, the asset units can be used to settle debts when both parties agree to that.
In practise this means there is more currency units available, because asset units and currencies running on top of NXT can appreciate faster than NXTs appreciate vs fiat. This makes large stakeholders buy some of those assets, thus spreading their large stakes among other members of the NXT economy.
NEM has a different approach. It tries to increase velocity of money by making people transact more often and with more important NEMsters to be able to harvest block fees. This approach has yet to prove itself. The NEM economy will have to grow very fast to give people options to spend their NEMs. Otherwise where would all those transactions come from?