To my understanding, in a PoS chain, both forks of a chain retain the staking balances on both sides, and it would only make sense for the stakers to validate the same blocks on both chains.
Sure, but can't the same apply on Proof-of-Work too? Say that the chain splits to old-Bitcoin and new-Bitcoin that supports merged-mining. Miners that mine the old chain can use redundant hashes to mine for the new chain, with the exact same hash rate. All that's needed for new-Bitcoin is to prove you've worked for it. It'll be essentially a sidechain, but with no dependence on the mainchain.
I don't see how Proof-of-Stake defeats the point of split. It's not the miners/stakers who define the value of the coins, but the users. If a split occurs, there might be new money created, but the product remains the same. Market value of 1 BTC is just split to 1-old-BTC and 1-new-BTC.
I don't think there ever was a Fork-BTC that was designed to be able to be mined merged with the original BTC. This seems like a very constructed example.
So, while in PoW model this seems like a constructed, rare exception, in PoS this is the
default. You can 'mine' / validate on both chains
by default, while in PoS you must decide where you point your miners and what chain you burn your electricity on.