The central issue with proof-of-stake is that when the chain forks, the stakeholders on the left side of the fork, with very few exceptions, are also stakeholders on the right side of the fork.  Whichever fork wins, they still have their stake.  In fact, they have an incentive to use it to mine both sides of the fork at once.  There's literally no rational reason for them not to; whichever fork eventually fails is eliminated from history, but until they know for sure which one that's going to be, why would they stop mining it?  And as a result of that dysfunction, why does a choice between forks ever get made at all?

Proof-of-work is the expenditure of a finite resource (hashing power) in real time.  You cannot use the same hashing power to support both sides of a fork; you are forced by physics to choose one side or the other, and therefore a decision on the local level perforce actually gets made - leading to a decision on the wider level.

The nothing-at-stake problem can be overcome by GHOST protocol or similar; it must not be possible to support one side of a fork without your repudiation or withdrawal of support being visible to the other.