Can someone explain the context of what Nakamoto exactly meant with: "To implement a distributed timestamp server on a peer-to-peer basis, we will need to use a proof- of-work system..."? I don't quite get the importance of the time aspect, since the timestamp header field seems not to be very important. Maybe someone can clearly outline, with real use cases and thought experiments, which exact problems we want to avoid to run into. Solana's proposed proof of history method (with Verifiable Delay functions) tries to exactly solve that problem. What I don't get: why is the passing of time the only thing we really want to prove? Concise problem descriptions are essential to understand the solution and caveats associated with it.
To put it simply: digital things are infinitely reproducible with very little cost, and so if we are attempting to create a digital money system, we need to make sure that users cannot simply copy & paste their money, to create (infinitely) more money. Functionally, this means that we need to solve the double-spend problem, and to do that, what we ultimately need is a way to give all transactions an explicit ordering. If we can reliably determine the order of transactions, then we can keep track of the network's state, and ensure that no user is allowed to spend their funds more than once.
With centralized systems, this is easy: you just trust the network admin to decide what the order is, and all other participants must follow suit.
But with a decentralized system... how do you create this ordering, in a way that all participants will reliably agree upon? The solution to this problem, as provided by Satoshi Nakamoto, is the bitcoin blockchain. It is a clock, with each block being the next 'tick' forward in the network's ongoing history.
This is the crucial point that allows the network to prevent users from double-spending their funds.
I don't quite get the importance of the time aspect, since the timestamp header field seems not to be very important.
The timestamp in the block header is more or less a sanity check, and it is used to help calculate network difficulty. More info can be found here.
This timestamp is not used to order transactions though, the blockchain itself is the 'clock' that provides that functionality.