What is the core problem that bitcoin solves? The distributed consensus problem.
There have been chains of hashes and chains of digital signatures before. What makes bitcoin different is that it is timestamping these digital messages, and protecting those timestamps against being reversed. The currency aspect of bitcoin is simply a layer on top of the distributed timestamping service. namecoin is an example of a non-currency use of distributed timestamping.
Actually I think you are oversimplifying a bit... just timestamping isn't enough. If all Bitcoin did was timestamp SHA256 hashes of transactions, it'd be useless because there would be still no consensus about double spends. The actual data itself must be public.
Better is to describe Bitcoin as an data store with inherently difficult to change order. Now it happens to be that Bitcoin is implemented in such a way that it will never store conflicting data twice, but if it wasn't, it would still work just fine - the second transaction would be useless garbage and ignored. Similarly actually having the
time attached to the data is only an artifact of the PoW implementation. Again, the order is what matters, not the time. If the difficulty was fixed at the current value, rather than adjusting every two weeks, Bitcoin would work just fine strictly speaking, though you would have to wait months to get a block.(1)
I'm being a bit pedantic... but it's an important distinction in the case of alt-chains that try to bootstrap off the Bitcoin PoW function. For instance you could try to define an alt-chain where ordering was determined by getting the block header hash timestamped into Bitcoin - I've proposed the idea before on the bitcoin-dev email list - but solving the actual consensus problem of being sure everyone knows about all transactions is very tricky with such "pseudo-mining"
1) You know, Bitcoin could have been usefully implemented via telegraph and semaphores back in the late 1800's had the block interval been set to, say, 1 month or so. A small team of human calculators could probably calculate a SHA256 hash for a transaction in a few hours, especially with purpose built mechanical aids, and back then they could have used a much weaker hash function and gotten away with it. Though they'd have eventually had a rousing debate about Sir. Charles Babbage's "mechanical work engines", not to mention allowing more than a few dozen transactions per month...