But the chains become independent once they split.
I now see it clearly. You're right. You can't use the same work to extend another chain, unless the new chain is dependent on the old chain, regardless of the configuration of the new chain. The new chain founders can either choose to completely disconnect it with the old chain, or make it
based on it.
Exactly, and imagine the credibility of a 'fork team' who on one hand is trying to sell their fork as the 'new Bitcoin' that will 'replace it' (
flippening), while at the same time being based (thus reliant) on the real BTC blockchain through merged mining. Nobody would take them seriously.
There is a binary decision which chain to mine, that doesn't exist in PoS.
That's also why there is a (trivial) binary decision which chain is longer in an unintentional chain split; meanwhile in Ethereum someone will need to decide which chain is 'longer', as the validators will just validate both sides of the split. Will it be based on a Tweet by Vitalik? I don't know. Doesn't inspire much confidence.
Forking the chain in the former fashion (e.g., Bitcoin Cash) in Proof-of-Stake does grant the validators the same benefits oppositely to Proof-of-Work, indeed. That being said, you could say that Proof-of-Work makes splitting / forking less desirable.
On the other hand, making it harder to fork also gives security / stability to Bitcoin, and if someone were to try coming up with a real contender, the economics (miners switching or not) will show which chain / which coin is more popular and should be pursued. In other words; it would be easier to figure out the 'winner'.