Do mining pools eliminate the need to store blockchain?
A mining pool does need to store the entire blockchain, however that doesn't mean that every participant of the mining pool needs to store it.  Someone in the pool needs it, but not everyone.

Individuals in a pool can just be sent block headers and perform the proof-of-work without ever verifying the transactions themselves.  This does not require the full blockchain.  However, someone should have already verified the transactions ahead of time, and this node does need the blockchain.