If the blockchain records all transactions, how does losing your wallet imply that you lose your Bitcoin? My understanding is that the blockchain records every transaction that has ever occurred, and that, when we say there are x amount of coins at address y, what we really mean is that the blockchain records transactions in which the difference between the total amount of Bitcoin going to that address and leaving that address is x.  In other words, it doesn't store the amount of Bitcoin at an address per se, but instead determines this by looking at transactions.

If my understanding of this is correct, then I don't understand how you are at risk of losing your Bitcoin if, for example, your hard drive crashes.  The blockchain would still be able to tell how much coin was at that address, so all you would need to be able to do is prove that that address belonged to you.

If the problem is that the address goes away with the client and (most likely) the user didn't have their address memorized, then why not just add the ability to access the same address with more than one client?  This way, if, for example, they lost their client on their desktop at home, they could still look up their address on their client at work, on their laptop, etc, and be able to recover the 'lost' coin.
You've answered your own question - "all you would need to be able to do is prove that that address belonged to you." To prove the address belongs to you you need the associated private key, which is secret and typically kept on your computer. If you lose your key you are no different than any other pretender to the address.

This is why you must back up your wallet file containing the keys, so should something happen to your computer you can still access the coins using your backup.

Remember, this is a decentralized digital currency. There is no clerk you can go to with some documents who can authorize that this is your address. You must prove ownership in a universally verifiable digital way - by using your private key to provide a cryptographic signature.