First a summary, then feedback.
A summary in my own words: Originally, it was proposed that a PoS-based gadget would finalize the PoW chain. In June 2018, this plan was abandoned in favor of Eth2. In December 2018, it was proposed to salvage the original finality gadget plans – Eth2 clients must be aware of recent Eth1 block hashes to decide on Eth1 to Eth2 validator deposits, but deciding on validator deposits may be interpreted as Casper votes, which may finalize the Eth1 chain. For this to work, Eth1 clients would have to change their fork-choice rule to monitor the beacon chain, perhaps as an Eth2 light client, so that they can honor the finalization of blocks.
Feedback: I believe that an earlier plan was to replace PoW with full PoS, not just finality. Of course, finalization alone brings great value and should be considered alone. But perhaps it would be wise to also consider further steps to replace PoW with PoS. This would solve the ASIC miner centralization problem, and allow reducing block rewards significantly.