>What value does the Ethereum protocol provide when a disagreement splits the network in two, without any recourse for either of the parties involved?

Don't give splits such importance. The nature of blockchains means that the network is being constantly forked into many chains at its tip. It doesn't matter which protocol is running on that blockchain.

>Which hard social/computer science problems are solved through a "consensus" algorithm that essentially constitutes denying the existence of those who disagree?

Existance is not being denied. The chain is still there. The problem is that the rest of the people, a majority of nodes, prefer another version of the chain.

>With Bitcoin, determining whether a fork is successful is very simple: are the tokens on its blockchain as liquid as those on the Bitcoin one? If they are, they are just as useful for transferring value as the "original" Bitcoin.

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>What metric would we use to determine whether an Ethereum fork is successful or not? If nothing is lost when a consensus algorithm fails to establish consensus, what value did it provide in the first place?

Proof of Work consensus algorithms don't "fail" to establish consensus. They are designed to swiftly choose one of many possible chains, based on a few simple parameters (the most important one being the amount of work needed to create the chain). That said, value is an off-chain concept. Just as US dollars have value because we all know they are widely accepted, Ethereum has value because there is a team of developers plus a big community plus a lot of nodes, and all that together allows some use cases that make Ethereum have value. Ethereum Classic has less value because it is not as widely accepted. Bitcoin has a lot of value because as of now it is the most popular cryptocurrency. Nothing about this has to do with the tech itself, but with the way we see the tech.